Ratios, Proportions, & Percents - SAT Math
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Portia purchased a laptop for $480, but after checking the merchant's website realized that she had been overcharged by 20%. By how much, in dollars, was she overcharged?
Portia purchased a laptop for $480, but after checking the merchant's website realized that she had been overcharged by 20%. By how much, in dollars, was she overcharged?
In any percent problem, it is crucial that you know exactly which value the percent is to be taken of. Here Portia is overcharged by 20%, meaning that she paid 20% more than the item should have cost. That means that the amount she paid, $480, equals 20% more than she should have paid:
, where 
is the price she should have paid.
To solve for 
, divide both sides by 
:
This means that she should have paid $400 but instead paid $480, so she overpaid by $80.
Note that a common mistake on percent problems is taking the percent of the wrong value: here you might be tempted to simply take 20% of 480, for example. The SAT test makers know that, and will often set up problems so that you're given the result of the percent change (here the result is Portia paying $480) and need to work your way back to the start value as you did here. Always pause to ensure that you know which value (or variable) gets multiplied by the percent change.
In any percent problem, it is crucial that you know exactly which value the percent is to be taken of. Here Portia is overcharged by 20%, meaning that she paid 20% more than the item should have cost. That means that the amount she paid, $480, equals 20% more than she should have paid:
To solve for 
This means that she should have paid $400 but instead paid $480, so she overpaid by $80.
Note that a common mistake on percent problems is taking the percent of the wrong value: here you might be tempted to simply take 20% of 480, for example. The SAT test makers know that, and will often set up problems so that you're given the result of the percent change (here the result is Portia paying $480) and need to work your way back to the start value as you did here. Always pause to ensure that you know which value (or variable) gets multiplied by the percent change.
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Over the course of a single week, the price of a company’s stock rose by 10%. If the following week the company’s stock rose by 10% of the new price, by what percent did the stock increase over the two week period?
Over the course of a single week, the price of a company’s stock rose by 10%. If the following week the company’s stock rose by 10% of the new price, by what percent did the stock increase over the two week period?
This problem rewards students who do the work rather than just looking at a problem and going with what “looks right.” Whenever you see a problem that asks you to find the percent difference between two unknowns, you can probably pick your own numbers to make the problem easier since the problem is asking for a relationship rather than a value.
In this case, since you are dealing with percents, it is a good idea to start with the number 100. If you increase 100 by 10%, you will get
after the first increase.
During the second week, you are told, the price of the stock increases by 10% of the new value. Be careful here to take the percent of the correct value, 110, rather than the original value. If you increase 110 by 10%, you get:
.
To find the percent increase from there, you find the difference and then divide by the original before multiplying by 100:
This problem rewards students who do the work rather than just looking at a problem and going with what “looks right.” Whenever you see a problem that asks you to find the percent difference between two unknowns, you can probably pick your own numbers to make the problem easier since the problem is asking for a relationship rather than a value.
In this case, since you are dealing with percents, it is a good idea to start with the number 100. If you increase 100 by 10%, you will get
During the second week, you are told, the price of the stock increases by 10% of the new value. Be careful here to take the percent of the correct value, 110, rather than the original value. If you increase 110 by 10%, you get:
To find the percent increase from there, you find the difference and then divide by the original before multiplying by 100:
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A store buys a shirt for $60, then marks it up by 50%, then offers customers a 20% discount off the marked up price. What price do consumers pay?
A store buys a shirt for $60, then marks it up by 50%, then offers customers a 20% discount off the marked up price. What price do consumers pay?
Remember that when you are dealing with taking the percent of a number and then taking the percent of the result, that you must be careful to take the percent of the correct number. In this case, that means increase $60 by 50% and then taking 20% off that new number. What you cannot do is either take off 20% of 60 or just increase 60 by 30% rather than doing each step in order. Doing either will lead to a trap that the Testmaker has left for you.
The first step you’re given is to increase $60 by 50%. This is the same as taking 150% of $60, or as multiplying it by 1.5. If you do this you get;
From there, you now need to find what 20% off 90 is in order to get your solution. You can do this by first finding 10% of 90 by moving the decimal point one place to the left to get 9 and then doubling it to get 18.
Subtracting 18 from 90 gives that a customer would pay $72.
Remember that when you are dealing with taking the percent of a number and then taking the percent of the result, that you must be careful to take the percent of the correct number. In this case, that means increase $60 by 50% and then taking 20% off that new number. What you cannot do is either take off 20% of 60 or just increase 60 by 30% rather than doing each step in order. Doing either will lead to a trap that the Testmaker has left for you.
The first step you’re given is to increase $60 by 50%. This is the same as taking 150% of $60, or as multiplying it by 1.5. If you do this you get;
From there, you now need to find what 20% off 90 is in order to get your solution. You can do this by first finding 10% of 90 by moving the decimal point one place to the left to get 9 and then doubling it to get 18.
Subtracting 18 from 90 gives that a customer would pay $72.
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Jenna's salary last year was 80% of what it is this year. By what percent did her salary increase from last year to this year?
Jenna's salary last year was 80% of what it is this year. By what percent did her salary increase from last year to this year?
With percent change problems it is important to recognize that you take the percentage of the original value. The formula is 
, and then multiply by 100 to convert to a percent.
Recognizing that, the percent change here is 
. This is the type of calculation you should be able to do quickly in your head: to go from 80 to 100, a number must increase by one-quarter of itself, which is 25%.
With percent change problems it is important to recognize that you take the percentage of the original value. The formula is 
Recognizing that, the percent change here is 
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Dan and Alex measure themselves against a tree in their backyard. The tree is 80% taller than Dan and 20% taller than Alex. Which of the following statements is true?
Dan and Alex measure themselves against a tree in their backyard. The tree is 80% taller than Dan and 20% taller than Alex. Which of the following statements is true?
This percents problem can be set up by noting the height of the tree (
) in two ways:
(the tree is 80% taller than Dan)
(the tree is 20% taller than Alex)
Now you can set the equations equal, since each defines the value of 
:
With this, the denominators cancel (you can just multiply both sides of the equation by 5), leaving:
Divide both sides by 
and you'll have your answer:
, which simplifies to: 
.
This means that Alex is 50% taller than Dan.
This percents problem can be set up by noting the height of the tree (
Now you can set the equations equal, since each defines the value of 
With this, the denominators cancel (you can just multiply both sides of the equation by 5), leaving:
Divide both sides by 
This means that Alex is 50% taller than Dan.
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At the end of 2007, a plot of land was worth 80% more than it had been worth at the beginning of 2006. If the land appreciated in value by 60% for the year 2006, by what percent did it increase for the year 2007?
At the end of 2007, a plot of land was worth 80% more than it had been worth at the beginning of 2006. If the land appreciated in value by 60% for the year 2006, by what percent did it increase for the year 2007?
A critical element of all percent problems is that you must pay particular attention to what the percent is being taken OF. Here, recognize that you're given two percents in terms of the beginning (start of 2006) value, but you're asked for the percent change for the year 2007, meaning that you're looking for the percent increase in terms OF the start of 2007 value.
If you say that the initial value at the beginning of 2006 was 
, then you know that at the end of 2006 / beginning of 2007 the value was 
, and at the end of 2007 it was 
. What you're being asked, then is the percent change from when it was at 
to when it was 
. The difference is 
, so you can use the calculation 
. This means that you're looking at a fraction of 
, which translates to 12.5%.
Alternatively, you could pick numbers to make this easier. If you say that the land was worth $100 to start, grew to $160 by the end of the first year, and then was worth $180 at the end of the second year, then you know that it gained $20 from a base value of $160 over the course of the second year, again for a 12.5% increase.
A critical element of all percent problems is that you must pay particular attention to what the percent is being taken OF. Here, recognize that you're given two percents in terms of the beginning (start of 2006) value, but you're asked for the percent change for the year 2007, meaning that you're looking for the percent increase in terms OF the start of 2007 value.
If you say that the initial value at the beginning of 2006 was 
Alternatively, you could pick numbers to make this easier. If you say that the land was worth $100 to start, grew to $160 by the end of the first year, and then was worth $180 at the end of the second year, then you know that it gained $20 from a base value of $160 over the course of the second year, again for a 12.5% increase.
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After the first week of training for an upcoming race, Dinesh begins the second week of his training program by increasing his weekly running mileage by 50%. After three weeks, he catches a cold and has to reduce his current weekly mileage by 40% for the final week of training. By what percent did Dinesh's weekly mileage change from the first week to the final week?
After the first week of training for an upcoming race, Dinesh begins the second week of his training program by increasing his weekly running mileage by 50%. After three weeks, he catches a cold and has to reduce his current weekly mileage by 40% for the final week of training. By what percent did Dinesh's weekly mileage change from the first week to the final week?
Algebraically, if Dinesh started at 
miles, he increases to 
miles in the second week, and decreases to 
for the final week -- a 10 percent decrease from his starting point.
Algebraically, if Dinesh started at 
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In a bakery, 70% of the doughnuts are glazed. If glazed doughnuts account for 21% of all pastries in the bakery, what percent of all pastries in the bakery are doughnuts?
In a bakery, 70% of the doughnuts are glazed. If glazed doughnuts account for 21% of all pastries in the bakery, what percent of all pastries in the bakery are doughnuts?
You can set this problem up algebraically by recognizing what you're really solving for, which is essentially a ratio of doughnuts (
) to total (
). You're given that the ratio of glazed (
) to doughnuts (
) is 70/100, and you're given that glazed (
) to total (
) is 21/100. Knowing that, you can set up:
and
And since 
has to be equal in each of those equations, you can set the equations equal:
From there, your goal is to solve for 
, so you'll divide both sides by 
to get:
And then multiply both sides by 
to cancel the fraction on the left and move all the numbers to the right. Now you have:
Now you should see how cleanly the terms on the right factor, leaving you with:
. If 
out of 
is 
out of 
, that means that the answer is 
.
You can set this problem up algebraically by recognizing what you're really solving for, which is essentially a ratio of doughnuts (
and
And since 
From there, your goal is to solve for 
And then multiply both sides by 
Now you should see how cleanly the terms on the right factor, leaving you with:
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During a clearance sale, a retailer discounted the original price of its TVs by 25% for the first two weeks of the month, then for the remainder of the month further reduced the price by taking 20% off the sale price. For those who purchased TVs during the last week of the month, what percent of the original price did they have to pay?
During a clearance sale, a retailer discounted the original price of its TVs by 25% for the first two weeks of the month, then for the remainder of the month further reduced the price by taking 20% off the sale price. For those who purchased TVs during the last week of the month, what percent of the original price did they have to pay?
With percent problems, the key is often to make sure that you take the percent of the correct value. In this case, the initial 25% off means that customers will pay 75% of the original price. Then for the second discount, keep in mind that the discount is taken off of the sale price, not of the original price. So that's 20% off of the 75% that they did pay, which can be made easier by looking at what the customer does pay: 80% of the 75% sale price. Using fractions, that means they pay: 
of the original price, which nets to 
of the original price, or 60%.
With percent problems, the key is often to make sure that you take the percent of the correct value. In this case, the initial 25% off means that customers will pay 75% of the original price. Then for the second discount, keep in mind that the discount is taken off of the sale price, not of the original price. So that's 20% off of the 75% that they did pay, which can be made easier by looking at what the customer does pay: 80% of the 75% sale price. Using fractions, that means they pay: 
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A part-time employee whose hourly wage was decreased by 20 percent decided to increase the number of hours worked per week so that the employee's total income did not change. By what percent should the number of hours worked be increased?
A part-time employee whose hourly wage was decreased by 20 percent decided to increase the number of hours worked per week so that the employee's total income did not change. By what percent should the number of hours worked be increased?
We can set up equations for income before and after the wage reduction. Initially, the employee earns 
wage and works 
hours per week. After the reduction, the employee earns 
wage and works 
hours. By setting these equations equal to each other, we can determine the increase in hours worked: 
(divide both sides by 
) 
We know that the new number of hours worked will be 25% greater than the original number.
We can set up equations for income before and after the wage reduction. Initially, the employee earns 
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is 
of half of a number. What is that number?
Here it is helpful to recognize that 
is the same thing as 
. So if 
is 
of something, it is 
of that thing, meaning you can multiply 
by 
to get 
. And we know that 
then is half of the number they’re looking for, so if we multiply by 
we have the number: 
.
Here it is helpful to recognize that 
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At a manufacturing factory, 
of all widgets are found to be defective. If the plant were to produce 
widgets, how many would be defective?
At a manufacturing factory, 
There are a couple of math shortcuts that can make this question much quicker than calculating the “long way” of multiplying 
times 
without a calculator.
Option 1 is to realize that “percent” means “out of one hundred.” 
defective means that 
out of every 
are defective; since 
, the total number of widgets, is 
of 
, you can then say that 
of 
are defective -- if 
out of 
are defective, then (
of 
) out of (
of 
) would be defective. This gives you the correct answer, 
.
Option 2 is similar: you can express 
as the fraction 
, which reduces to 
. If you then know that 
out of every 
widgets is defective, and you have a total of 
widgets, which is 
times 
, you’ll have 
defective widgets.
There are a couple of math shortcuts that can make this question much quicker than calculating the “long way” of multiplying 
Option 1 is to realize that “percent” means “out of one hundred.” 
Option 2 is similar: you can express 
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What is 
increased by 
?
What is 
To increase 
by 
, you’d use the expression 
. But instead of doing the decimal multiplication, you can express 
as 
. That makes your job significantly easier, using:
Since 
equals 
, if you take 
of that you’ll just have 
. Add 
to the original 
and you have the correct answer, 
.
To increase 
Since 
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What is 
decreased by 
?
What is 
A decrease of 
is the same thing as a decrease of 
, since 
, which reduces to 
. A convenient way to handle percent decreases is to think that if 
has been taken away, then 
is left. So you can multiply 
by 
to get 
.
A decrease of 
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Tamara spent 
of her 
-hour workweek in meetings. How many hours that week was she NOT in meetings?
Tamara spent 
To calculate 
efficiently, you can first calculate 
by just moving the decimal point one place to the left. That means that 
of 
is 
(or just 
). Then to get 
, take half of what you had as 
. Since 
here is 
, then 
is half that: 
. Add those together, and you have 
. Here that means that 
is equal to 
. If she worked 
total hours and 
were in meetings, then the other 
were not in meetings.
To calculate 
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If 
of a number is 
, what is that number?
If 
A great way to perform a calculation like this is to set up a proportion. If you know that 
, you can then cross-multiply and solve for 
. (That proportion means that 
, 
out of 
, is equal to 
out of some other number.) Solving, you’d have:
Simplify the first fraction: 
Cross-multiply: 
Divide by 
to isolate 
: 
The correct answer is 
.
A great way to perform a calculation like this is to set up a proportion. If you know that 
Simplify the first fraction:
Cross-multiply:
Divide by 
The correct answer is 
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If 
of a number is 
, what is that number?
If 
A great way to set up a percent calculation like this is as a proportion:
That translates to 
(
out of 
) is equal to 
out of an unknown number.
You can then simplify the fraction 
to 
to make the calculation faster. If you then have:
You have two options. One, you can recognize that the difference between the numerators is that the second fraction is twice the second, and you can then say that the denominator must also be double, giving you 
, meaning 
.
Or you can cross-multiply and solve, giving you:
A great way to set up a percent calculation like this is as a proportion:
That translates to 
You can then simplify the fraction 
You have two options. One, you can recognize that the difference between the numerators is that the second fraction is twice the second, and you can then say that the denominator must also be double, giving you 
Or you can cross-multiply and solve, giving you:
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is 
of a number. What is that number?
A great way to calculate this type of percentage problem is by setting up a proportion:
This translates to 
(
out of 
) is equal to 
out of an unknown number. When you then cross-multiply, you’ll see that the math sets up nicely for quick calculation:
Since 
is 
, you can likely do this math in your head to find that 
, giving you the correct answer, 
.
A great way to calculate this type of percentage problem is by setting up a proportion:
This translates to 
Since 
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Keith completed a 
-kilometer race, during which he walked for 
of the distance and ran the rest. How many kilometers did he run?
Keith completed a 
If Keith walked 
of the distance, that means he ran 
of the distance. So you can calculate 
of 
. 
can be expressed as 
, which reduces to 
, making the calculation by hand much quicker. 
of 
is 
, making 
the correct answer.
If Keith walked 
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If 
is 
of 
, what is 
?
If 
One great way to set up a percent calculation like this is to follow the language of the prompt. The word “is” means equals, and “per cent” means “divided by 
,” so this prompt literally translates to the equation:
You can then reduce the fraction 
by factoring out a 
from both numerator and denominator, leaving:
Then if you multiply both sides by 
, you’re just about ready to solve:
Since 
divided by 
is 
, you can then multiply 
to get to 
, the correct answer.
One great way to set up a percent calculation like this is to follow the language of the prompt. The word “is” means equals, and “per cent” means “divided by 
You can then reduce the fraction 
Then if you multiply both sides by 
Since 
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