Basic Single-Variable Algebra

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Math › Basic Single-Variable Algebra

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1

An art show wants to focus on sculpture but will also accept paintings and drawings. The show has room for pieces and will display them in a ratio of scupltures to painting to drawings. How many sculptures will be displayed?

Explanation

Since the paintings and the drawings have the same ratio, they will have the same number on display.

Let number of paintings displayed = number of drawings displayed.

Then, number of sculptures displayed.

Set up a proportion:

Solve by cross multiplying to get or .

Then add to both sides to get .

Then divide by to get .

So there are drawings and paintings in the show, leaving the last spots for sculptures.

2

Tom is painting a fence feet long. He starts at the West end of the fence and paints at a rate of feet per hour. After hours, Huck joins Tom and begins painting from the East end of the fence at a rate of feet per hour. After hours of the two boys painting at the same time, Tom leaves Huck to finish the job by himself.

If Huck completes painting the entire fence after Tom leaves, how many more hours will Huck work than Tom?

Explanation

Tom paints for a total of hours (2 on his own, 2 with Huck's help). Since he paints at a rate of feet per hour, use the formula

(or )

to determine the total length of the fence Tom paints.

feet

Subtracting this from the total length of the fence feet gives the length of the fence Tom will NOT paint: feet. If Huck finishes the job, he will paint that feet of the fence. Using , we can determine how long this will take Huck to do:

hours.

If Huck works hours and Tom works hours, he works more hours than Tom.

3

In April, the price of a t-shirt is . In May, the store increases the price by 50%, so that the new price is . Then in June, the store decreases the price by 50%, so that the t-shirt price is now . What is the ratio of to ?

Explanation

If the original price of the T-shirt is , increasing the price by 50% means that the new price is 150% of , or .

If the price is then decreased by 50%, the new price is 50% of

or

The ratio of to is then:

The 's in the numerator and denominator cancel, leaving , or

.

4

Solve the equation for .

No solution

Explanation

Since we have variables on both sides, we first subtract from both sides.

This leaves us with .

Subtracting 2 from both sides, we get .

5

An art show wants to focus on sculpture but will also accept paintings and drawings. The show has room for pieces and will display them in a ratio of scupltures to painting to drawings. How many sculptures will be displayed?

Explanation

Since the paintings and the drawings have the same ratio, they will have the same number on display.

Let number of paintings displayed = number of drawings displayed.

Then, number of sculptures displayed.

Set up a proportion:

Solve by cross multiplying to get or .

Then add to both sides to get .

Then divide by to get .

So there are drawings and paintings in the show, leaving the last spots for sculptures.

6

In April, the price of a t-shirt is . In May, the store increases the price by 50%, so that the new price is . Then in June, the store decreases the price by 50%, so that the t-shirt price is now . What is the ratio of to ?

Explanation

If the original price of the T-shirt is , increasing the price by 50% means that the new price is 150% of , or .

If the price is then decreased by 50%, the new price is 50% of

or

The ratio of to is then:

The 's in the numerator and denominator cancel, leaving , or

.

7

Sarah notices her map has a scale of . She measures between Beaver Falls and Chipmonk Cove. How far apart are the cities?

Explanation

is the same as

So to find out the distance between the cities

8

Sarah notices her map has a scale of . She measures between Beaver Falls and Chipmonk Cove. How far apart are the cities?

Explanation

is the same as

So to find out the distance between the cities

9

Solve the equation for .

No solution

Explanation

Since we have variables on both sides, we first subtract from both sides.

This leaves us with .

Subtracting 2 from both sides, we get .

10

Tom is painting a fence feet long. He starts at the West end of the fence and paints at a rate of feet per hour. After hours, Huck joins Tom and begins painting from the East end of the fence at a rate of feet per hour. After hours of the two boys painting at the same time, Tom leaves Huck to finish the job by himself.

If Huck completes painting the entire fence after Tom leaves, how many more hours will Huck work than Tom?

Explanation

Tom paints for a total of hours (2 on his own, 2 with Huck's help). Since he paints at a rate of feet per hour, use the formula

(or )

to determine the total length of the fence Tom paints.

feet

Subtracting this from the total length of the fence feet gives the length of the fence Tom will NOT paint: feet. If Huck finishes the job, he will paint that feet of the fence. Using , we can determine how long this will take Huck to do:

hours.

If Huck works hours and Tom works hours, he works more hours than Tom.

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