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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 147 miles. If you use 227 as an approximation for π, 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: a=Fm. Suppose F=713 and m=15. Find a, 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: a=223÷15.

223×51=1103=3623. Therefore, the acceleration is 3623 m/s2.

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

Use the formula speed=distancetime.

s=1.5÷14

Rewrite the expression and simplify: s=3241=4321=122=6 mi/hr.

  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:

3÷12=3×21=3×2=6.

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

2÷6=2×16=26=13.

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 147 miles and 227 is used as an approximation for π.

The formula for the circumference is C=πd where d is the diameter. If the circumference is 147, we need to divide this by π to find the diameter. 

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

147=1×7+47=117 

Now, divide:

147÷227=117÷227=117×722=77154=12

Therefore, the diameter of the race track is 12 of a mile.

Example 2

Newton’s Second Law relates acceleration to the force of an object and its mass: a=Fm. Suppose F=512 and m=23. Find a, the acceleration.

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

512=5×2+12=112

a=Fm=11223=112÷23=112×32=11×32×2=334=814

Therefore, the acceleration is 814m/s2.

Example 3

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

Use the formula speed=distancetime:

speed=distancetime=314÷12=134÷12=134×2=13×24.

Before we continue, we will simplify the fraction:

13×24=13×22×2=132=612.

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

Review

In 1 – 3, evaluate the expression.

  1. xy for x=38 and y=43
  2. 4z÷u for u=0.5 and z=10
  3. 6m for m=25
  4. The label on a can of paint states that it will cover 50 square feet per pint. If I buy a 18-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 38 mph. How long will it take to dig a trench 23-mile long?
  6. A 27 Newton force applied to a body of unknown mass produces an acceleration of 310 m/s2. Calculate the mass of the body. Note: Newton=kg m/s2
  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. 199(11)
  2. \begin{align*}-2.3 - (-3.1)\end{align*}
  3. \begin{align*}|16-84|\end{align*}
  4. \begin{align*}|\frac{-11}{4}|\end{align*}
  5. \begin{align*}(4 \div 2 \times 6 + 10-5)^2\end{align*}
  6. Evaluate \begin{align*}f(x)= \frac{1}{9} (x-3); f(21)\end{align*}.
  7. Define range.

Review (Answers)

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

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Vocabulary

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

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