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# Applications of Reciprocals

## Express all values as fractions. Divide by multiplying by the reciprocal

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Applications of Reciprocals

Suppose that a car did one lap around a circular race track with a circumference of miles. If you use as an approximation for , could you find the diameter of the race track?

### Using Reciprocals to Solve Real-World Problems

The need to divide rational numbers is necessary for solving problems in physics, chemistry, and manufacturing.

#### Let's use reciprocals to answer the following problems:

1. Newton’s Second Law relates acceleration to the force of an object and its mass: . Suppose and . Find , the acceleration.

Before beginning the division, the mixed number of force must be rewritten as an improper fraction.

Replace the fraction bar with a division symbol and simplify:

. Therefore, the acceleration is

1. Anne runs a mile and a half in one-quarter hour. What is her speed in miles per hour?

Use the formula .

Rewrite the expression and simplify:

1. For a certain recipe of cookies, you need 3 cups of flour for every 2 cups of sugar. If Logan has 1/2 cup flour, how many cups of sugar will he need to use to make a smaller batch?

First we need to figure out how many times bigger 3 is than 1/2, by dividing 3 by 1/2:

Since 1/2 goes into 3 six times, then we need to divide the 2 cups of sugar by 6:

Logan needs 1/3 cup of sugar to make a smaller batch with 1/2 cup flour.

### Examples

#### Example 1

Earlier, you were asked to find the diameter of a race track if the circumference is  miles and  is used as an approximation for .

The formula for the circumference is  where  is the diameter. If the circumference is , we need to divide this by  to find the diameter.

First, we have to turn the mixed number into an improper fraction:

Now, divide:

Therefore, the diameter of the race track is  of a mile.

#### Example 2

Newton’s Second Law relates acceleration to the force of an object and its mass: . Suppose and . Find , the acceleration.

Before we substitute the values into the formula, we must turn the mixed fraction into an improper fraction:

Therefore, the acceleration is .

#### Example 3

Mayra runs 3 and a quarter miles in one-half hour. What is her speed in miles per hour?

Use the formula :

Before we continue, we will simplify the fraction:

Mayra can run 6-and-a-half miles per hour.

### Review

In 1 – 3, evaluate the expression.

1. for and
2. for and
3. for
4. The label on a can of paint states that it will cover 50 square feet per pint. If I buy a -pint sample, it will cover a square two feet long by three feet high. Is the coverage I get more, less, or the same as that stated on the label?
5. The world’s largest trench digger, “Bagger 288,” moves at mph. How long will it take to dig a trench -mile long?
6. A Newton force applied to a body of unknown mass produces an acceleration of . Calculate the mass of the body. Note:
7. Explain why the reciprocal of a nonzero rational number is not the same as the opposite of that number.
8. Explain why zero does not have a reciprocal.

Mixed Review

Simplify.

1. Evaluate .
2. Define range.

To see the Review answers, open this PDF file and look for section 2.11.

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### Vocabulary Language: English Spanish

reciprocal

The reciprocal of a nonzero rational number $\frac{a}{b}$ is $\frac{b}{a}$.