This video will show you ACT Math Strategies to help you with the ACT Test. If you need help, find an ACT Math Tutor below.
ACT Math Strategies [Video Transcript]
A motorcycle manufacturer offers three different models, each available in six different colors. How many different combinations of model and color are available? There are a couple of different ways we could figure out this problem. First, I’m going to show you a visual representation, so if we say our models are A, B and C, these are the three different models of cars you can get. And then for colors we use the numbers one through six and so there are six colors. Then you could get model A in color one, color two, color three, color four, color five or color six.
Same thing with model B, you could get model B in color one, color two, color three, color four, color five or color six. And finally the same with model C, color one, color two, color three, color four, color five and color six. And if we count up all of these different combinations, we have 6 for model A, 6 for model B, and 6 for model C, or 6 times 3 which is 18. So instead of writing this all out, we could have taken our six colors and multiplied it by our three models. Three times 6 is 18, so there are 18 different model and color combinations.
Find the length of the side labeled X. The triangle represented in the figure is a right triangle, as shown. To find the missing side of any right triangle, we can use the Pythagorean theorem which is A squared plus B squared equals C squared, where the A and the B are the legs of the right triangle, or the two sides that form the right angle in the right triangle, and C is the hypotenuse of the right triangle, or the side that’s across from the right angle. So now we need to substitute the values from our triangle to the Pythagorean Theorem.
Your A could either be 15 or X because both of those are legs. So I’m going to use 15, 15 squared plus, and I could replace the B with X, X squared equals, C again is the hypotenuse just 25 squared. Now we need to simplify by squaring 15 and 25, 15 squared means 15 times 15 which is 225, plus X squared is equal to 25 squared which is 25 times 25 or 625. Now we’re trying to find what X is which means we need to get X alone, to do that, the first thing we need to do is subtract 225 from both sides, 225 minus 225 is zero, bring down the X squared term, bring down the equal sign, and 625 minus 225 is 400 and we’re almost done.
Again, we’re trying to find X, not X squared, so in order to solve for X, we need to do the opposite of squaring X. The opposite of squaring X is square rooting X, so we must square root both sides to solve for X. The square root of X squared is simply X, equals the square root of 400 is 20, since 20 times 20 is 400. And when you square root a number, you’re really asking a question. And the question is what number times itself is 400? That answer is 20, so our answer is B.
An airplane leaves Atlanta at 2:00 p.m. and flies north at 250 miles per hour. A second airplane leaves Atlanta 30 minutes later, and flies north at 280 miles per hour. At what time will the second airplane overtake the first? Well this is a dirt problem or distance equals rate times time. You see, we’ve got time, 30 minutes later we’ve got rate, 280 miles per hour, 250 miles per hour. We’ve got distance here because these planes are flying a distance and that’s how their rate and time are related, so we need to use this formula to help us organize our data.
So first let’s find the distance for the first airplane. The rate of the first airplane is 250 miles per hour, so 250 times, and then all we know is what time the first airplane left. We don’t know how long that airplane flew, so we’ve got to use T for time there. Now the distance for the second airplane, the rate is 280 miles per hour, and we do know a little bit about the time that the second airplane flew. If you’ll notice, the second airplane is going faster than the first one. So if they left from the same place and they go in the same direction, they’re both going north.
Then the second airplane is eventually going to catch up with the first and actually pass the first, and that’s what we’re trying to figure out is at what time will the second airplane catch up to, or overtake the first? So the second airplane won’t have been flying as long as the first airplane when it catches up. It’s going faster, and it left 30 minutes later. So however long this first airplane has been flying, the second one will have been flying that time minus 30 minutes since it left 30 minutes later. Now if you think about this, you’ve got two planes leaving from Atlanta, they’re both flying north and the question is when will the second one overtake the first one?
Well this is what we’re trying to find out is what time that will happen. Well at that time, both of these airplanes will have flown the exact same distance. We don’t know what that distance is but we know that it’s the same, so we can take these two distances and we can set them equal. So the distance of plane one 250T is equal to the distance of plane 2, 280 times the quantity T minus 30. And now we need to solve for T. So the first thing we want to do is distribute our 280, so I’m going to bring down this 250T equals 280 times T, 280T minus 280 times 30, 280 times 30, 0 place holder, 3 times 0 is 0, 3 times 8 is 24. Three times 2 is 6, plus 2 is 8, that’s 8,400.
Now we need to get all of our variables on the same side of the equation. We don’t want to have T on the left side and T on the right. So I’m going to subtract 280T from both sides, 250T minus 280T. I’m taking away more than I have, so I’m going to end up with a negative, and the difference between 280T and 250T is 30T, so negative 30T equals, and by subtracting this 280T, I’ve actually cancelled that term on the right side. And I’m left with negative 8,400. To solve for T, then we need to divide both sides by negative 30. Negative 30 divided by negative 30 is 1 times T is simply T.
Negative 8,400 divided by negative 30, first of all, a negative divided by a negative is a positive. And then we can cancel a 0, and really all we have to do is divide 840 by 3. Really I’m running out of space here. I’m going to get rid of this, so we can divide 840 by 3. Three goes into eight two times, that’s six. Subtract and you get 2, bring down the 4, 3 goes into 24 8 times. Three times 8 is 24 subtract and you get 0.
Bring down your zero, three goes into zero, zero times. So it’s 280, but 280 what? Well if you’ll recall, we did T minus 30 minutes, so that would mean that these Ts are in minutes, so we just found how many minutes the first plane, T, the first plane would have been flying when the second plane overtook it. So we need to first convert these minutes into hours and minutes, so that we can then find the time, what time it was when that happened. So to convert these minutes into hours, there are 60 minutes in one hour, which means we need to divide 280 by 60, so you’ve got 60, 120, 180, 240.
So 60 goes into 280, 4 times and you get 240. You subtract and you get a remainder of 40 minutes, so that’s 4 hours 40 minutes. So 4 hours 40 minutes after the first plane left at 2:00 p.m. is when the second plane overtook it, or caught up with it, or had travelled the same distance. So 2 plus 4 hours is 6 and 40 minutes. So at 6:40 is when that second plane overtook the first. The answer is C.
Mark is driving to Phoenix from a city located 210 miles north. He drives the first 10 miles in 12 minutes. If he continues at the same rate, how long will it take him to reach his destination? We can use a proportion to solve this problem. We know that the first 10 miles took him 12 minutes, so that’s 1 ratio, 10 miles took 12 minutes. What we want to know is how long it took him to drive the 210 miles, and since it’s at the same rate, he continues in the same rate, we can use this proportion to solve and find how long it’s going to take him in minutes.
Miles to minutes, miles to minutes, and to solve a proportion, we cross multiply, so 10 times X is 10X equals 12 times 210. So when I multiply that right over here. Two times zero is zero, two times one is two, two times two is four. Zero place holder, one times zero is zero. One times one is one. One times two is two. And we add these together, 0, 2, 5, 2, so 2520, and then to solve for X, we need to divide both sides by 10, so X equals, and just cancel 0 here, 252. Two hundred fifty two, what?
Well 252 minutes, but as you can see, our answers aren’t in minutes, they’re in hours and minutes, so we need to figure out just how many hours and then left over minutes there are in 252 minutes. And since there are 60 minutes in one hour, I need to divide 252 by 60, so 252 divided by 60. So 60 does not go into 25, 60 goes into 252, 60, 120, 180, 240, so four times because 4 times 6 is 24, and 4 times 60 is 240. We subtract and we get 12 that’s left over, so that means 252 minutes is 4 hours and 12 minutes left over. So that is answer choice B, 4 hours and 12 minutes.
If A equals negative 6 and B equals 7, then 4A times the quantity 3B plus 5 plus 2B equals? If we want to find the value of this expression, then the first thing we need to do is substitute negative six and seven for A and B respectively. So four…this is actually four times A, any time you see a number and a letter written next to each other like this, that’s multiplication. So I know that when I replace A with a negative six, it’s going to read four times negative six. And you can use a little dot for a times sign.
Or you can even use the X for the times sign. I’m just going to use parentheses, and then these parentheses here, since there’s no sign between the A and the parentheses. Not a plus, not a minus sign or a division sign. Again, it’s understood to be multiplication, so this is times the quantity three. Again, times B, B is seven plus five, plus two again, times B, which is seven. Now there are a lot of parentheses here, so what we can do to minimize confusion is we can change these parentheses into brackets, and it still means the same thing.
They act exactly like parentheses do, but sometimes it makes it look a little less confusing if you don’t have all those parentheses everywhere. So in order to simplify this expression, we have to follow the order of operations, sometimes referred to as PEMDAS. I’m going to write that over here just so we have it to refer to, which is that we first always muse simplify inside our parentheses first, and we would simplify any terms with exponents next, followed by simplifying our multiplication and division operations from left to right, in our expression.
So just whichever one occurs first in our expression from left to right. And finally we would simplify our addition and subtraction operations, also from left to right. So following PEMDAS, we first need to simplify these expressions inside our grouping symbols, so I’ll bring down my four times negative six, and then inside my grouping symbols, I also have to follow the order of operations. So now looking inside these grouping symbols, I have two different operations. I have multiplication, three times seven, and addition, plus five.
So if we refer back to PEMDAS, we need to do our multiplication before we do our addition, so I’m going to start by multiplying 3 times 7 which is 21, and I’ll bring down plus 5 plus 2 times 7. We haven’t finished simplifying inside our parentheses, or grouping symbols, so we must continue simplifying inside our grouping symbols until we’re done. So that means the next thing I’m going to do is simplify 21 plus 5, so I have 4 times negative 6 times the quantity 26. And I can now just use parentheses again, because I’m just multiplying times another number. Plus two times seven.
So we’ve finished with the parentheses which really I know it looks like we haven’t, but what this parentheses, what that P actually means is not just parentheses. Any parentheses you do simplify, no, no, no, doesn’t mean that. It means any parentheses that act as grouping symbols. So parentheses that group more than one term together. And if you’ll look at our parentheses, none of the ones we have up here are grouping more than one term together. Each set of parentheses just has a term in it, because it’s really acting as a multiplication sign, not as a grouping symbol in this case.
So the only operations we have left are multiplication 4 times negative 6 times 26, and addition. And again, according to PEMDAS, we need to do that multiplication before we get to addition. And we work multiplication division simply from left to right. So I’ll first multiply 4 times negative 6, which is negative 24. Positive times a negative is a negative. Times 26 plus 2 times 7. We want to continue working our multiplication from left to right until we’ve simplified all multiplication, which means I need to multiply negative 24 times 26 next.
And I’m going to do that just right here to the side, 26 times 24. Four times 6 is 24 so you write the 4 and carry the 2. Four times 2 is 8 plus 2 is 10, and then we use a 0 place holder so we can move on to the 2. Two times 6 is 12, write the 2 carry the 1. Two times two is four, plus one is five. Four plus 0 is 4, 0 plus 2 is 2, 1 plus 5 is 6, 624, and that’s after we added, but it was a positive 26 times a negative 24, which means it’s a negative 624 plus 2 times 7. We’re almost there. We need to continue with our multiplication until we’ve multiplied all the terms there are to multiply, which means we next need to multiply our 2 times 7, so we have negative 624 plus 2 times 7 is 14.
Now we’ve finally simplified it down to our very last operation, and that is addition, and we’re adding two numbers with different signs. One is negative, one is positive. When you’re adding numbers with different signs, you need to subtract the numbers and take the sign of the number with the larger absolute value. So 624 minus 14 is 610, and since 624 is negative, our answer is negative. So your answer is D, negative 610.
Which of the following is a solution to the inequality 4X minus 12 is less than 4? First, we should solve for X in this inequality, so we’re going to add 12 to both sides, bring down our 4X minus 12 plus 12 is 0 so that cancels. Bring down our less than sign and 4 plus 12 is 16. Then we need to divide both sides by 4, 4 divided by 4 is 1 times X is X is less than 16 divided by 4 is 4. Keep in mind when solving inequalities that if you multiply or divide both sides by a negative number, you must flip the inequality symbol.
In this case we didn’t have to since we didn’t divide both sides by a negative number. It’s all about that negative. So now we can apply easy answer choices. We want to find which one of these numbers is less than four, is seven less than four? No. Is six less than four? No. Is five less than four? No. What about four? Is four less than four? No. If this were a less than or equal to sign that said, “X is less than or equal to four, then yes, four would be a solution because four is equal to four. And it only has to be one, either less than or equal to.
But in this case it’s not a less than or equal to sign. It’s simply a less than sign. Meaning that this number must be smaller than four. So our answer is E, three is less than four, for three is a smaller number than four.
What is the area of the figure shown? Well there are a few different ways you can approach this problem, but I’m going to show you one, and that is to first find the area of this whole rectangle and then simply to subtract the area of this rectangle here, since that’s the only part that’s missing out of this large rectangle. So first we’ll find the area of the large rectangle. And the area of a rectangle is the length times the width, so the area of the large rectangle is 20 feet times 50 feet, so the area is 2 times 5 is 10 0 0, 1,000 feet times feet, feet squared.
So the area of the large rectangle is 1,000 feet squared. Now we need to find the area of this smaller rectangle. The only part that’s missing out of our large rectangle, so I’m going to label this, this is the area of the large rectangle. Now we’re going to find the area of the small rectangle. So again area is the length times the width. And we have the length and the width for our small rectangle as well, 15 feet and 6 feet. So the area is 15 feet times 6 feet. Six times 10 is 60. Six times 5 is 30. So 30 plus 60 is 90. Feet times feet is feet squared. So the area of this little rectangle is 90 feet squared. So to find the area of this figure, we have to subtract the area of our large rectangle, 1,000 feet squared minus the area of that small rectangle cut out which is 90 feet squared. And 1,000 minus 90 is 910 feet squared, and that’s the area of our figure.
Lauren had $80 in her savings account. When she received her paycheck, she made a deposit which brought the balance up to one $120. By what percentage did the total amount in her account increase as a result of this deposit? This is a percent increase problem, and there’s a formula for that. The formula to find a percent of change is to find the amount of change, divide that by the original amount and then multiply that amount by 100. And the reason you multiply by 100 is because when you divide, you’ll get decimal answer, and to change a decimal to a percent, you multiply by 100.
Or simply move the decimal two places to the right. So first we need to find our amount of change. The amount of change is found by subtracting the two amounts, so she started at $80 and her balance went up to $120, so the change is one 120 minus 80, divided by the original amount. So what she had in her account before she made her deposit, the $80, and then we multiply that by 100. So the percent change is 120 minus 80, 40 divided by 80 times 100. To simplify this, we can first cancel zero, so we have four-eighths times 100, and four-eighths can be simplified to one half.
Or since we want to change it to a percent, five-tenths which again is equivalent to four-eighths or one half. So we multiply that times 100. This is our decimal, but we want a percent, so multiplying by 100, simply moves your decimal two places to the right. And we have to fill this empty seat with a zero, so it’s 50%. It was a 50% increase, answer A.
The distance traveled by a moving object is computed from the relation distance equals rate times, time. Where R is the rate of travel, speed and T is the time of travel. A major league pitcher throws a fastball at a speed of 125 feet per second. The distance from the pitching rubber to home plate is 60.5 feet. How long in seconds does it take a fastball to travel this distance? Compute your answer to the nearest hundredth of a second.
So first I’m going to pull out this formula that’s going to be very helpful in solving this problem. Distance equals the rate times the time. Now I’m going to take the information I was given and substitute it into this equation, so they told me the speed of the ball, and remember speed is your rate R, so that means I’m going to replace R with 125.
Then they also gave me the distance 60.5 feet, which means I’m going to replace D with distance, 60.5, and so T is what I’m trying to find. So the T will stay there in my equation. So in order to find T, to find time, we need to solve for T by dividing both sides by 125, 125 divided by 125 is 1, times T is T, so to find our answer, we need to divide 60.5 by 125. So I’ll work it over here, 60.5 divided by 125. So that decimal goes right there, 125 does not go into 60, but it does go into 600.5 so you’ve got 125, 250, 500 and then one more time would be 625.
So that means it only goes four times. And that is 500 and we subtract and we get 105. Add another zero bring it down. We want to see how many times 125 goes into 1050. Well if you remember, that 125 times 4 was 500, then it would get twice as many times as that, so 8 times, and that would give us 1,000. We subtract that and we get 50 add another 0, bring it down, and 125 goes into 500 4 times, 125 times 4 is 500 subtract and you get 0. So that means that the time would be 484 thousandths seconds.
But if we go back to our instructions, it said to compute your answer to the nearest hundredth, which means I need to round this to the hundredth place, the eight is in the hundredth place and that four tells the eight to stay the same. So it’s simply 48 hundredths seconds.
Elijah drove 45 miles to his job in an hour and 10 minutes in the morning. On the way home however, traffic was much heavier and the same trip took an hour and a half. What was his average speed in miles per hour for the round trip? So first let’s determine just how far Elijah went. It was 45 miles to his job and then he had to drive 45 miles back home. So he drove that 45 miles twice or 90 miles. Speed is calculated by dividing your distance by your time.
So we’ve got his total distance. Now we need to find his total time. It says it took him an hour and 10 minutes to get to his job, so one hour and 10 minutes is a total of 60 minutes plus 10 minutes, 70 minutes. Then on the way home it took him an hour and a half, so 1 hour 30 minutes. Again an hour 60 minutes plus 30 minutes means a total of 90 minutes. I’m converting the time into minutes just because it make those units easier to work with, when you’ve got hours and minutes together, it’s easier just to bring them all into minutes. Put them all into that same unit of minutes.
So 70 plus 90 is 160 minutes for his total travel time, so again to calculate his speed, we need to divide his total distance by his total time, so 90 miles divided by 160 minutes. Well the first thing I could do to simplify this is I could cancel a zero. So now it’s just 9 divided by 16, which will be a lot easier to work with. So I have 9 divided by 16, 16 doesn’t go into 9 at all. That’s 0 which means I’m going to need to add a decimal and a 0, 16 goes into 90 5 times. So 16 times 5, 5 times 6 is 30. Carry the 3, 5 times 5 plus 3 is 8 so that’s 80. We subtract and we get 10.
So we need to add another zero and bring it down. So 16 goes into 100, 6 times since it went into 16 times 5 was 80, then 1 more time would be 96, and then we subtract and we get 4. Add another 0 bring it down, 16 goes into 40, 2 times, 16 times 2 is 32, so we subtract and we get 8. Add another 0 bring it down, and 16 goes into 80, 5 times, 16 times 5 is 80. We subtract and get 0, so that means that his speed in miles per minute would 0.5625, but they didn’t ask us for his speed in miles per minute.
They asked us for his average speed in miles per hour, so now I have to convert this speed into miles per hour by multiplying it by 60, since there’s 60 minutes in 1 hour. So if he travels this speed every minute, then we multiply by 60 to find the speed in miles per hour. So first start with your zero place holder. Six times 5 is 30, so that’s 0 carry the 3, 6 times 2 is 12 plus 3 is 15, 6 times…put that in the wrong place oops, 6 times 6 is 36 plus 1 is 7. Carry the three. Six times 5 is 30 plus 3 is 33, and we have 1, 2, 3, 4 numbers behind the decimal. So your answer should have four numbers behind the decimal. So it’s 33 and 75 hundredths miles per hour, which is the same as 33 and three fourths miles per hour since three fourths and 75 hundredths are equal.
The diagram shows the outline of a racetrack for skaters, which consists of two long straight sections and two semicircular turns. Given the dimensions shown, which of the following most closely measures the perimeter of the entire track? Well they’ve given us that this length here is 150 yards apparently since all of our answers are in yards, so that means that this opposite straight side here would also be 150 yards, and actually just from that amount of information, we can already start eliminating some answers.
I know that 180 yards is definitely not enough, because I’ve got 150 and 150, so we’re going have to skate this 150, skate some more, skate another 150, and skate some more, so we know we’re already over 180 yards. We also know it’s going to be more than 300 yards because just the two straight sections are 300 yards, so we can already eliminate that answer choice, and we’re down to these three possibilities. So that means we need to figure out what this distance is here and here. They told us that these were semicircular turns. That means that semi, semi means half of a circle.
So if you take these two halves of a circle and you put them together, you get one whole circle. So really they’re travelling the distance of one whole circle in addition to these straight sides here, which means we need to find the circumference of our circle to find the distance around the circle. That’s what circumference is, and the circumference of a circle is pi times the diameter of the circle, which they gave us. They told us that this distance is 30. Well that goes all the way across my circle, so that’s my diameter, so my circumference is 30 pi.
And pi is about three. It’s actually a little over three but since they said most closely measures, we’d probably be okay just using three for pi. So our circumference is about 90 yards, and plus we have these straight sides so the total distance around the race track would be the 150 yards plus 150 yards for our 2 straight sides, plus about 90 for these 2 curved sides, the semicircular which is three 390 yards. Now if you’ll remember, we rounded pi down. It’s actually over three. We rounded it down to three so really our answer is a little bit higher than 390 which is answer choice D, 395 yards.
Morris started work today at 7:00 a.m. and worked until 4:30 p.m. He earns $12 per hour for his regular shift which is 8 hours and 50% more per hour for overtime. How much did Morris make today? The first thing we want to do is find out how long Morris worked, so if he worked from 7:00 a.m. to 4:30 p.m. and then from 7:00 a.m. to noon, that’s 5 hours, and from noon to 4:30 p.m. is four and a half hours, which means he worked for a total of nine and a half hours. Now they told us that he makes he makes $12 an hour for just his regular 8 hour shift, so for 8 of these hours, he made $12 an hour, and the rest, the leftovers, so one and a half hours, would be his over time.
And for overtime he makes 50% more per hour, so his regular pay would be 8 hours times $12 an hour, which is $96, but then for overtime he makes one and a half times as much as he made for his regular shift. So 12 times 50% more which would be one and a half times he works at for one and a half hours. So first I’m going to multiply 12 times one and a half. Five times 2 is 10 carry the 1. Five times 1 is 5, plus 1 is 6, put a 0 place holder.
One times two is two. One times 1 is 1, so you add this together and we get 180, but there’s 1 number behind the decimal, so that means that he was making $18 an hour for that one and a half hours that he worked overtime. So now we multiply 18 times one and a half, and 5 times 8 is 40, carry the 4. Five times one is five plus four is nine. Zero placeholder, one times eight is eight. One times one is one. We add those together, 17 carry the 1, 1 number behind the decimal. One number behind the decimal.
So that means he made $27 in overtime, so his total amount that he made that day would be the sum of the amount of money he made for his regular shift, and the amount he made for overtime so we need add $96 and $27. Seven plus 6 is 13 carry the 1, 9, 10, 11, 12, so that means he made $123 that day.
Pradip decides to invest $4500 in Cisco Systems stock and buys it at the price shown in the table. At what price should he sell it to obtain a profit of 10%? So he is buying Cisco systems stock at $3.50 per share. If he wants to make a profit of 10%, then he needs to sell his stock when the price per share is 10% higher, and there are several ways to solve this problem. I’ll show you one, so first I want to know what is 10% of $3.50.
Well of means to multiply, so I’m multiplying times one tenth or times 10 hundredths. I’m multiplying times 10 hundredths means just moving that decimal one place to the left. So that is 35 hundredths or 35 cents, so that means that his stock needs to be 35 cents higher. The price per share needs to be 35 cents greater. So $3.50 plus 35 cents is $3.85, so in order to make a profit, a 10% profit, then he needs to sell his stock at $3.85 per share.
Marjorie buys a package of stocks consisting of 100 shares each of Microsoft and Apple, as well as 200 shares of Garmin at today’s closing prices, as shown in the table. What is the average price per share that she pays for these stocks? To find this average price per share, first I need to find the total price. Then I have to find the total shares and divide because this is a ratio, price per share. Price divided by shares, so first I’m going to figure out how much money she spent on these shares, so she got 100 shares of Microsoft. So let’s start with Microsoft.
She bought 100 shares for $45.14 each, so times 45.14, and multiplying by 100 just means that you move that decimal 2 places to the right, so it’s $4,514 that she spent on her Microsoft shares. She also bought 100 shares of Apple. so 100 times an Apple of $16.90, and again multiplying by 100 just means to move that decimal 2 places to the right, so she spent $1,690 on her Apple shares. Then she got 200 shares of Garmin, so Garmin 200 times. And Garmin is $29.30. So multiplying by 100 means we move it 2 place to the right.
But multiplying by 200 means not only do we multiply our…or move our decimal two places to the right, but we’re also going to double that number. So if you double zero you get zero. If you double three you get six. If you double 9 you 18, and then we carry the 1 over here. Two times two is four plus one is five. So all I did was multiply 200 times $29.30. So I can use this then to find my total amount of money spent. I’m just going to move this up a little so I have some more room.
So I’m going to add these three costs together to find my total cost, or my total price. So four plus zero plus zero is four. One plus 9 is 10 plus 6 is 16, and then we have 8 plus 6 is 14, plus 5 is 19 plus 1 is 20. Five plus 1 is 6 plus 4 is 10, plus 2 is 12. So that means she spent a total of $12,064 on all of these stocks. That’s our price. Now we need to find our shares and that total can be found right here, just add these up, 200 plus 100 plus 100 that’s a total of 400 shares, so the average price per share would be $12,064 divided by 400 shares.
So all we have to do is divide those 2 numbers, 400 goes into 1206 3 times, 400 times 3 is 1200, subtract and you get 6 bring down the 4. Four hundred does not go into 64 at all, so that’s a 0 which means we add a decimal and a 0 and bring it down. Four hundred goes into 640 1 time, so bring that decimal up. One times 400 is 400, then you subtract and you get 240. Add another zero bring it down, and 400 goes into 2400 6 times, 400 times 6 is 2400. You subtract and you get a reminder of 0, which means the average price per share $30.16.
David bought 200 shares of Oracle stock yesterday and sold it today. His profit was $22. At what price did he buy the stock yesterday? We have this table of information to use, so we’re only focusing on Oracle stock which presently is being sold for $19.11 per share. it tells us that he bought 200 shares and made a profit of $22, so the first thing we want to do is find out his profit per share, which means we need to divide his profit by his number of shares. Two hundred does not go into 22, so I have to add a decimal and a 0.
Now 200 will divide into 220. Two hundred goes into 220 1 time and 200 times 1 is 200. We subtract and we get 20, so we add another 0 and bring it down, 200 goes into 200 one time, 200 times 1 is 200. We subtract and we get zero. Now we know that he made 11 cents per share. If today’s price per share is $19.11, then we have to subtract his profit to find out how much he bought them for the day before. So $19.11 minus 11 cents would just be $19, answer B.
The figure shows an irregular quadrilateral and the lengths of its individual sides, which of the following equations best represents the perimeter of the quadrilateral? The perimeter is the distance around a figure. To find the perimeter of any shape you can simply add all of the sides, so the perimeter is M plus three plus M plus 2, plus M plus 2M, and now we just need to combine like terms. So the perimeter is M plus M plus M plus 2M, so that’s 1, 2, 3 Ms plus 2Ms is a total of 5Ms, plus 3 plus 2 is 5. So the perimeter is 5M plus 5, answer D.
Figure nine shows two quarter circles centered on the origin of the Cartesian coordinate plane. The inner circle has a radius of two units, the outer circle has a radius of three units. What is the area of the shaded region? So that’s right in here, so if you can just imagine it, this circle would continue going as would this larger circle. So since it’s been split into quarters, all we’re finding the area of is a quarter of those circles. So first let’s start with what the formula for the area of a circle is, its pi times radius squared.
As we discussed though this is only a quarter of the circle, so our area is a quarter of the area of these circles, so and then we need to find the area of just the part between those two circles, so that would be the area of the large circle minus the area of the small circle. The area of the large circle would be pi times the radius which is three squared minus the area of the small circle, which would be pi times two squared, and the three and the two came from the radius. The radius of the small circle is two and the radius of the outer circle has a radius of three.
So that’s where these numbers came from. And now let’s simplify that, so we get area is a fourth of three squared is nine so nine pi minus, two squared is four, so four pi. Now since each one of these terms has a pi in it, we can factor the pi out. So we get area equals a fourth pi times the quantity nine minus four. And then we can simplify nine minus four, so we have area equals a fourth pi times five and a fourth times five is five fourths so that’s five fourths pi. But of course if you’re needing to bubble this in on an answer sheet, then you don’t have the option of bubbling pi, so we’re going to have to multiply five fourths times pi.
And five fourths times pi is 1 in 25 hundredths pi, and for pi we’ll just use 3 and 14 hundredths so we’re doing one in 25 hundredths times 3 and 14 hundredths, so that I’m going to do over here. So 3 and 14 hundredths times one and 25 hundredths, and that’s my pi. Five times 4 is 20. Five times 1 is 5, that’s 7, 5 times 3 is 15. Put a zero place holder. Two times four is eight, two times one is two, two times three six. Put two zero place holders. One times four is four, one times one is one. One times three is three. Add that together, that’s zero. Seven plus 8 is 15 carry the 1, 5, 9, 10, 12 carry the 1, 6, 7, 8, 9, 3. And we then we have four numbers behind the decimal one, two, three, four numbers behind the decimal. So and then we could round this to 3 and 93 hundredths would be the area of just the shaded region.
Francine can ride 16 miles on her bicycle in 45 minutes. At this speed, how many minutes would it take Francine to ride 60 miles? This problem is a good candidate for using a proportion to solve. There’s more than one way to solve it, but I like proportions so I’m going to use a proportion. First I know that Francine can go 16 miles in 45 minutes, that’s going to be my first ratio. So I have 16 miles in 45 minutes, and the key to setting up a successful proportion is to be consistent. So since our first ratio we have miles to minutes.
Then our second ratio will also be miles to minutes. So they want to know how many minutes it would take Francine to ride 60 miles, so my miles is 60 and my minutes is what I’m trying to find so that’s where I’ll put a variable. To solve a proportion we cross multiply, 16 times X is 16X and that equals 60 times 45, 0, 6 times 5 is 30 carry the 3. Six times 4 is 24 plus 3 is 27, so 2700. And then we solve for X by dividing both sides by 16, so X equals, and now I need to find out what 2700 divided by 16 is.
Sixteen goes into 2700 1 time, and that’s 16, and we subtract and get 11. Then we bring down our zero. We could use 15 as a compatible number to see how many times 15 would go into 110, so we have 15, 30, 45, 60, 75, 90, 105, so 7 times. So let’s see if 16 will go into 110 7 times. Seven times 6 is 42, 7 times 1 is 7 plus 4 is 11. And that’s a little too big which means I know 16 goes into 110 6 times, so 16 times 6, 6 times 6 is 36. Six times one is six plus three is nine. And that works not so we need to subtract and we get 14.
Then we bring down our next zero. So we have 140. So I know 16 is going to go 140 at least 7 times, but it should go even more than that, so let’s see what 16 times 8 is. Eight times 6 is 48, 8 times 1 is 8 plus 4 is 12. And that’s all we’re going to be able to do so 8 times, and we get 128. And when we subtract those numbers we get 12. So that means we need to add a decimal and a zero, and bring it down, 16 goes into 120 7 times, 16 times 7 we found earlier is 112. So we subtract and we get eight. We add another zero and bring it down, and 16 goes into 80 less than 6 times and actually exactly 5 times. Sixteen times 5 is 80, so we subtract and we get a remainder of 0. So that means if it took her 45 minutes to go 16 miles, it should take her 168.75 minutes to go 60 miles, and this is the answer you would grid in.
In an election in Kimball County, candidate A obtained 36,800 votes. His opponent, candidate B, obtained 32,100 votes, 2100 votes when to write in candidates. What percentage of the votes went to candidate A? To find a percent, we need the part over the whole, and we know the part. The part number of votes candidate A received, which is 36,800. What I need to know is the whole or the total number of votes. To find the total number of votes, we need to add up 36,800, 32,100 and 2100 so that we can find the total.
So we’re adding. Those are zero and again we get zero. Then we have 8, 9, 10, so that’s a zero carry the 1. We have 6, 7, 8, 9, 10, 11 carry the 1, and we have 3, 6, 7, so that means there were a total of 71,000 votes. Now to find our percent, we need to divide, and that’ll give us our decimal which will then change into a percent. But before I divide, I’m going to first simplify this fraction by canceling two zeros, and now my division will be a lot easier. So we need to divide 368 by 710. Well 710 doesn’t go into 368 at all, which means I’m going to have to add decimal and a zero.
And now I can divide 710 into 3680, so I can use compatible numbers here and 710 is very close to 700, 700 times 5 would be 3500. So I’m thinking 710 is going to go about 5 times into 3680. So I’m going to multiply this, we get 0, and that’s 5, and that’s 35 so 3550. Yep, looks good, so that’s 5. We have 3550, now I need to subtract and that’s 0, 3, 1, so 130. Then we need to add a 0 and bring it down 1300, so 710, and it’s only going to go into 1300 1 time because 700 plus 700 would be 1400 and that’s too much.
So it’s just 1 time, 710 times 1 is 710, then we need to subtract. So I’m going to borrow from the 1, that’s a 0, that’s 13 borrow from the 13, that’s 12 which makes that 10, and we’ll borrow…well we don’t need to borrow that 0, so 10 minus 1 is 9 and 12 minus 7 is 5, 590 which makes sense. If you add 90 back to 710, you get 800, and if add 500 to 800, you get the 1300. So we need to add another zero and bring it down. So now we need to know how many times 710 goes into 5009. So I’m again going to use that compatible number 700.
So let’s see 700 times 8 would be 5600, so I’m thinking it goes in there about 8 times. So that’s 0 and that’s 8 and that’s 56. So yes, that’s very close so that’s 8 and we have 5680. We can subtract and that’s 0 borrow from the 9 that’s an 8 so that’s 10, 10 minus 8 is 2. Eight minus 6 is 2, so we get 220. And we could keep going but really that’s all I need to know because as I look at my answers, it’s very clear which one of these is my answer.
This is, well it’s about 518 thousandths and then when we change it into our percent, we move our decimal two places to the right, we get 51.8% which is answer A.
Jesse invests $7000 in a certificate of deposit that pays interest at the rate of 7.5% annually. How much interest in dollars does Jesse gain from this investment during the first year that he holds the certificate? This is a great problem to use this formula on, and some people call it IPRT just as a way to remember it. The I stands for the interest. The amount of interest in dollars like what we’re trying to find. P stands for principal or the amount that’s being invested initially, that would be our $7,000 the R is the rate or the percentage but written as a decimal.
So we’re going to take 7.5% and write it as a decimal and that will be our rate. And then T stands for time, so you may want to make a note of that this I is the interest P is the principle, R is the rate as a decimal, and the T is the time. So we’re going take our information from our problem and plug it in. So our amount of interest is equal to the principal, $7000, times the rate as a decimal again, so we’re taking 7.5% and changing it to a decimal, which means taking the decimal and moving it 2 places to the left. So you add that 0 so it’s 75 thousandths and then times the year or the time, and we’re just trying to find about the first year.
So that means it been there for one year and then we just need to multiply those things together. So one times anything is just that, so I can really ignore that, so I just need to multiply these two numbers together. So I have this 75 thousandths and I’m going to put it on top, because it’s very easy to multiply times 7000 since the first thing I’m going to do is just write down those 3 zeros. And then the only number I really have to multiply by is the 7, so 7 times 5 is 35. Write the five carry the three. Seven times 7 is 49, plus 3 is 52, and then we have 3 numbers behind the decimal, so 3 numbers behind the decimal and that means that he’s going to make $525 off of his investment.
So that’s the number you would need to bubble in is 525.
In the figure, A, B, and C are points on the number line, where O is the origin. What is the ratio of the distance BC to the distance AB? So first we need to find the distance between points B and C and the distance between two points is the absolute value of the difference of the coordinates, so that is the absolute value of B is five minus C is eight, which is the absolute value of negative three and that is three. So the distance between points B and C is three. Now we’re going to do the same thing to find the distance between points A and B.
And again, it’s the absolute value of the difference of the coordinates of point A and point B. So that’s the absolute value of A is negative six minus B is five. Which is the absolute value of negative six minus five is negative 11 and that is 11, so the distance from point A to point B is 11. Now they’ve asked us what the ratio of those distances is. And it gives you the order that it wants the ratio written in, so the distance BC should be the first number in the ratio three, two and we use a colon or you can write two is fine too.
The distance from points A to B which is 11, so our ratio is three to 11 or answer D.
Rachel spent $24.15 on vegetables. She bought two pounds of onions, three pounds of carrots and one and a half pounds of mushrooms. If the onions cost $3.69 per pound and the carrots cost $4.29, what is the price per pound of mushrooms? So first I want to figure out how much money was spent on the onions that were $3.69 per pound, and how much was spent on the carrots that were for $4.29 per pound. So let’s start with the onions. Those onions cost $3.69 for every pound that’s purchased, and she purchased two pounds.
So we need to multiply $3.69 times 2 to find the total amount of money spent on onions. So 2 times 9 is 18 carry the 1, 2 times 6 is 12 plus 1 is 13, write the 3 carry the 1, 2 times 3 is 6 plus 1 is 7. Two numbers behind the decimal, two numbers behind the decimal. So that means she spent $7.38 out of her total on onions. We’re going to do something similar to find the total amount of money spent on carrots. So next we’ll deal with our carrots. The carrots were $4.29 per pound so $4.29 times 3 since she bought 3 pounds of carrots. Three times 9 is 27. Write the seven carry the two.
Three times 2 is 6 plus 2 is 8, and 3 times 4 is 12. Two numbers behind the decimal, two numbers behind the decimal. so she’s spent a total of $12.87 on carrots, and so now I’m going to find the total spent on onions and carrots, so that means I’m just going to add these two values together. So $12.87 plus $7.38, make sure that when you add or subtract decimals that you line those decimals up. So we can go ahead and bring that down right now. Seven plus 8 is 15, 8, 9, 10, 11, 12, 7, 9, 10, so that means she spent $20.25 on the onions and carrots. And knowing that will help us find out what amount was left to be spent on the mushrooms.
Since I know her total spent on all of the vegetables was $24.15. I can take that total and subtract $20.25 which was spent on onions and carrots, and what I’ll have is the amount of money she spent on the mushrooms. So again we line up our decimals, five minus five is zero. I can’t take two from one so I borrow from the four it becomes a three. Then I have 11 minus 2 which is 9. Bring that decimal down three minus zero is three. So she had $3.90 to spend on mushrooms, but what we need to know is what the price per pound was, which we can find using this information.
She bought one and a half pounds of mushrooms, so we need to take the total amount spent on mushrooms and divide that by how many pounds of mushroom she purchased, which was one and a half, or one and five tenths. Now we need to move the decimal one place in both of our numbers, so that’s where it is now. And 15 goes into 39 2 times, 15 times 2 is 30 subtract and you get 9. Bring down the zero, 15 goes into 90 6 times, 15 times 6 is 90, subtract and you get zero, so no remainder, but we can add a zero here since we’re talking about money. So that means that the mushrooms were $2.60 per pound, which is answer A.
Mrs. Patterson’s classroom has 16 empty chairs. All the chairs are occupied when every student is present. If two fifths of the students are absent, how many students make up her entire class? This is a great problem to use a proportion to solve. We can take this fraction, two fifths, and use it as our first ratio. Two being the amount of students who are absent out of the five total. And that equals again number of students who are absent divided by the total, and it says in the problem that there are 16 empty chairs which means there are 16 students missing since usually all the chairs are occupied when every student is present.
So every empty chair is a student who’s absent. So we have 16 absent students out of, and this is what we don’t know. What’s the total, how many students make up her entire class? And to solve a proportion, we simply need to cross multiply 2 times X is 2 X and that equals 16 times 5 which is 80, and then we solve for X by dividing both sides by 2, and X equals 40, so that means there are 40 students in her entire class.
A sailboat is 19 meters long. What is its length in inches? There are many ways to approach this problem and solve it. The first thing I’m going to do is convert my meters into centimeters because then it’s a quick conversion into inches from there. So I know that one meter is equal to 100 centimeters, and we can use a proportion to solve. We can take this conversion factor and use it as a ratio, so one meter is 100 centimeters which equals 19 meters is how many centimeters. And then to solve the proportion, we cross multiply.
One times X is X, and 100 times 19 is 1900, so it’s 1900 centimeters. Now I can use another proportion to find inches. I know that 2.54 centimeters is 1 inch, and I have 1900 centimeters, and I need to know how many inches that is. And again, to solve a proportion, we cross multiply. So 2.54 centimeters times X, 2.54X is equal to 1 times 1900, which is 1900, and I forgot to put my little line up here, so I’ll do that. And now we need to solve for X which means dividing both sides by 2.54, 2.54 divided by 2.54 is 1 times X is X, and that equals 1900 divided by 2.54.
So we need to do some division, which means I have to move this decimal two places to the right. So I have to take my decimal of 1900 and move it 2 places to the right, which means adding 2 zeros. So now my number is 190,000 being divided by 254, so 254 does not go into 1 or into 19 or into 190, but it does go into 1900. And we can use compatible numbers here to get a very quick estimate of how many times it goes into 1900. So 254 is very close to 250 so 250, 500, 750, 1000, 1250, 1500, 1750, 1800…sorry 1750 and then 2000.
So it only goes 7 times because 2000 is too much. So 254 only goes 7 times into 1900. And we can see what that is right here, 7 times 4 is 28, 7 times 5 is 35, 36, 37. Seven times 2 is 14 plus 3 is 17. So it’s 1778. Then we subtract those, and we have to borrow from the 9 and that becomes an 8 so this is 10 and we borrow from the 10, it becomes a 9. This is 10, so 10 minus 8 is 2. Nine minus 7 is 2, and 8 minus 7 is 1, 122. So now we bring down our next zero.
And again, we can use that 250 rule to see how many times it goes into 1,220 so 250, 500, 750, 1000, and then it would be 1250, but 1250 is too much, so it only goes 4 times. So let’s see, 254 times 4, 4 times 4 is 16 carry the 1. Four times 5 is 20 plus 1 is 21. Four times 2 is 8 plus 2 is 10 so that’s 1016, and then we subtract. So we need to borrow from this 2, that becomes a 1, so this 10 minus 6 is 4, 1 minus 1 is 0, and 2 minus 0 is 2. So that’s 204, and then we bring down our 0 here.
So again, how many times does 250 go into to 2040? So we have 250, 500, 750, 1000, so 4 more times for 2000. So it looks like a total of eight times. So let’s check that. See where I have some room. I’ll just do that here, 254 times 8, 8 times 4 is 32, 8 times 5 is 40 plus 3, 43. Eight times 2 is 16, 16 plus 4 is 20, so 8 times is correct and it’s 2032. We subtract and we get 8 and that would be our remainder, it’d be 8 out of 254, but we could keep going to find our decimal. But if we look at our answers, we don’t need to keep going because none of our answers have the decimal as part of it, so our answer is D, 748 inches.
Jamie had $6.50 cents in his wallet when he left home. He spent $4.25 on drinks and $2 on a magazine. Later his friend repaid him $2.50 that he had borrowed the previous day. How much money does Jamie have in his wallet? So the first thing I’d like to do is find the total amount of money that Jamie spent, and I see that he spent $4.25 and $2. So to find the total amount of money he spent, I need to add $4.25 plus $2, and it’s very important that whenever you add numbers with decimals, or subtract numbers with decimals that you line up the decimals.
So now I’m ready to add, and I can just bring my decimal down before I even start, five plus zero is five, two plus zero is two, four plus two is six. So now I know he spent a total of $6.25. And if he spent that much money, then that’s money that is taken away from what he had. So he started with $6.50, but he spent $6.25. So I have to subtract what he spent from what he had before, and again we need to line up our decimals. I can’t take five from zero, so I’m going to borrow from this five. It becomes a four. And now I have 10 minus 5 is 5, and 4 minus 2 is 2. You can bring down that decimal, six minus six is zero.
So that means after he spent the $4.25 on the drinks and $2 on the magazine, he only had 25 cents left. But his friend repaid him $2.50, which means his friend gave him $2.50, so that’s money that’s added to his total. So now I’m going to take what he had after he spent his money, and add the money his friend gave him, and again we line up our decimals. Five plus zero is five, two plus five is seven. Bring down the decimal, zero plus two is two, so after spending money and getting repaid money, he now has a total of $2.75 which is answer D.