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? After completing this Concept, you'll be able to solve real-world problems such as this by using reciprocals.
Guidance
Using Reciprocals to Solve Real-World Problems
The need to divide rational numbers is necessary for solving problems in physics, chemistry, and manufacturing. The following example illustrates the need to divide fractions in physics.
Example A
Newton’s Second Law relates acceleration to the force of an object and its mass: . Suppose and . Find , the acceleration.
Solution: 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
Example B
Anne runs a mile and a half in one-quarter hour. What is her speed in miles per hour?
Solution: Use the formula .
Rewrite the expression and simplify:
Example C
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?
Solution: 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.
Video Review
Guided Practice
1. Newton’s Second Law relates acceleration to the force of an object and its mass: . Suppose and . Find , the acceleration.
2. Mayra runs 3 and a quarter miles in one-half hour. What is her speed in miles per hour?
Solutions:
1. Before we substitute the values into the formula, we must turn the mixed fraction into an improper fraction:
Therefore, the acceleration is .
2. Use the formula :
Before we continue, we will simplify the fraction:
Mayra can run 6-and-a-half miles per hour.
Practice
In 1 – 3, evaluate the expression.
- for and
- for and
- for
- 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?
- The world’s largest trench digger, “Bagger 288,” moves at mph. How long will it take to dig a trench -mile long?
- A Newton force applied to a body of unknown mass produces an acceleration of . Calculate the mass of the body. Note:
- Explain why the reciprocal of a nonzero rational number is not the same as the opposite of that number.
- Explain why zero does not have a reciprocal.
Mixed Review
Simplify.
- Evaluate .
- Define range.