# SAT Mathematics : SAT Math

## Example Questions

### Example Question #1 : Extrapolating Linear Proportions

A lawn can be mowed by  people in  hours. If  people take the day off and do not help mow the grass, how many hours will it take to mow the lawn?

Explanation:

The number of hours required to mow the lawn remains constant and can be found by taking the original  workers times the  hours they worked, totaling  hours. We then split the total required hours between the  works that remain, and each of them have to work  and  hours:  .

### Example Question #61 : Sat Math

What is  decreased by ?

Explanation:

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 .

### Example Question #62 : Sat Math

If  of a number is , what is that number?

Explanation:

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:

### Example Question #3 : Calculating Percents

is  of a number. What is that number?

Explanation:

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, .

### Example Question #4 : Calculating Percents

Keith completed a -kilometer race, during which he walked for  of the distance and ran the rest. How many kilometers did he run?

Explanation:

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.

### Example Question #5 : Calculating Percents

is  of half of a number. What is that number?

Explanation:

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: .

### Example Question #2 : Calculating Percents

At a manufacturing factory,  of all widgets are found to be defective. If the plant were to produce  widgets, how many would be defective?

Explanation:

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.

### Example Question #6 : Calculating Percents

What is  increased by ?

Explanation:

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, .

### Example Question #3 : Calculating Percents

Tamara spent  of her -hour workweek in meetings. How many hours that week was she NOT in meetings?

Explanation:

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.

### Example Question #7 : Calculating Percents

If  of a number is , what is that number?

Explanation:

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