### All SAT Math Resources

## Example Questions

### Example Question #1 : How To Find The Percent Of Increase

Cindy is running for student body president and is making circular pins for her campaign. She enlarges her campaign image to fit the entire surface of a circular pin. After the image is enlarged, its new diameter is 75 percent larger than the original. By approximately what percentage has the area of the image increased?

**Possible Answers:**

75%

200%

20%

225%

100%

**Correct answer:**

200%

Pick any number to be the original diameter. 10 is easy to work with. If the diameter is 10, the radius is 5. The area of the original image is A = πr^{2}, so the original area = 25π. Now we increase the diameter by 75%, so the new diameter is 17.5. The radius is then 8.75. The area of the enlarged image is approximately 77π. To find the percentage by which the area has increased, take the difference in areas divided by the original area. (77π - 25π)/25π = 51π/25π = 51/25 = 2.04 or approximately 200%

### Example Question #2 : Percent Of Change

If the side of a square is doubled in length, what is the percentage increase in area?

**Possible Answers:**

**Correct answer:**

The area of a square is given by , and if the side is doubled, the new area becomes . The increase is a factor of 4, which is 400%.

### Example Question #4 : Percent Of Change

A stock for YUM was trading at . If the price increased by , then decreased by , then increased by ; what was the net % change in price (to the nearest tenth of a percent)? (use a negative sign to denote a decrease)

**Possible Answers:**

**Correct answer:**

We will use the formula to solve this one

compute the terms in the parentheses:

If we rewrite the term in parentheses to match the form of the original formula, we can find the rate without having to do extra computation.

So, the rate is a decrease by 0.784%, which we round to 0.8%

### Example Question #4 : Percent Of Change

A circle has its radius increased by . By what percent is its area increased?

**Possible Answers:**

Cannot be determined with the information given.

**Correct answer:**

If we use *r* to denote the original radius of the circle, then according to the formula:

the new radius *R*, is given by

Therefore, the new area is:

Or

Since (pi)*r*^{2} is the area of the original circle, the rate of the increase is 21%

### Example Question #1 : Percent Of Change

At a certain store, prices for all items were assigned in January. Each month after that, the price was 20% less than the price the previous month. If the price of an item was *x* dollars in January, approximately what was the price in dollars of the item in June?

**Possible Answers:**

0.81*x*

0.51*x*

0.33*x*

0.66*x*

0.12*x*

**Correct answer:**

0.33*x*

The question tells us that the price in January was *x*. To find the price in February, we decrease the price of *x *by 20%, which is the same as taking 80% of *x.* (In general, a P% decrease of a number is the same as (100 – P)% times that number). Continue to take 0.8 times the previous month's price to find the next month's price until we have the price for June, as follows:

January Price: *x*

February Price: 0.8 * January Price = 0.8*x*

March Price: 0.8 * February Price = 0.8 * 0.8*x* = 0.8^{2 }*^{ }*x*

April Price: 0.8 * March Price = 0.8 * 0.8 * 0.8*x* = 0.8^{3 }*^{ }*x*

May Price: 0.8 * April Price = 0.8 * 0.8 * 0.8 * 0.8*x* = 0.8^{4 }*^{ }x

June Price: 0.8 * May Price = 0.8 * 0.8 * 0.8 * 0.8 * 0.8*x *= 0.8^{5 }*^{ }*x*; therefore, the price in June was 0.8^{5 }≈ 0.328 ≈ 0.33 times the original price.

### Example Question #1 : Percent Of Change

The sale of a tablet decreased from $500 to $450. By what percentage did the cost decrease?

**Possible Answers:**

10%

5%

1%

20%

15%

**Correct answer:**

10%

Set up the following ratio

50/500 = *n*/100

The cost of the tablet decreased by $50. The original cost is $500; therefore, 50 is the numerator and 500 is the denominator on the left side of the ratio.

Since a percentage is a part of a whole, *n* symbolizes the the percentage decrease.

To solve for *n*, you can cross multiply. So, 50(100) = *n*(500).

*n* = 10%

### Example Question #1 : Percent Of Change

On Monday, the price of a shirt costs *x* dollars. On Tuesday, the manager puts the shirt on sale for 10% off Monday's price. On Wednesday, the manager increases the price of the shirt by 10% of Tuesday's price. Describe the change in price from Monday to Wednesday.

**Possible Answers:**

1% decrease

1% increase

10% increase

No change

10% decrease

**Correct answer:**

1% decrease

To find the cost on Tuesday, take 10% off Monday's price. In other words, find 90% of Monday's price. This is simply 0.9*x*. If we are to now add 10% of this value back onto itself to find Wednesday's price, we want 100% + 10%, or 110% of 0.9*x*.

1.1(0.9*x*) = 0.99*x*

This value is 1% smaller than *x*.

### Example Question #2 : Percent Of Change

If the length of a rectangle is increased by thirty percent, which of the following most closely approximates the percent by which the rectangle's width must decrease, so that the area of the rectangle remains unchanged?

**Possible Answers:**

30

23

17

25

21

**Correct answer:**

23

### Example Question #1 : How To Find The Percent Of Decrease

If the price of a TV was decreased from $3,000 to $1,800, by what percent was the price decreased?

**Possible Answers:**

20%

30%

50%

40%

60%

**Correct answer:**

40%

The price was lowered by $1,200 which is 40% of $3,000.

### Example Question #41 : Percentage

If a rectangle's length decreases by fifteen percent, and its width decreases by twenty percent, then by what percent does the rectangle's area decrease?

**Possible Answers:**

40

36

35

45

32

**Correct answer:**

32

Let's call the original length and width of the rectangle and , respectively.

The initial area, , of the rectangle is equal to the product of the length and the width. We can represent this with the following equation:

Next, let and represent the length and width, respectively, after they have been decreased. The final area will be equal to , which will be equal to the product of the final length and width.

We are asked to find the change in the area, which essentially means we want to compare and . In order to do this, we will need to find an expression for in terms of and . We can rewrite and in terms of and .

First, we are told that the length is decreased by fifteen percent. We can think of the full length as 100% of the length. If we take away fifteen percent, we are left with 100 – 15, or 85% of the length. In other words, the final length is 85% of the original length. We can represent 85% as a decimal by moving the decimal two places to the left.

= 85% of =

Similarly, if we decrease the width by 20%, we are only left with 80% of the width.

= 80% of =

We can now express the final area in terms of and by substituting the expressions we just found for the final length and width.

= ()() =

Lastly, let's apply the formula for percent of change, which will equal the change in the area divided by the original area. The change in the area is equal to the final area minus the original area.

percent change = (100%)

=(100%)

=(100%) = –0.32(100%) = –32%

The negative sign indicates that the rectangle's area decreased. The change in the area was a decrease of 32%.

The answer is 32.