Problem 43
Question
A pair of basketball shorts requires \(\frac{3}{4}\) yd of nylon. How many pairs of shorts can be made from 24 yd of nylon?
Step-by-Step Solution
Verified Answer
32 pairs of shorts can be made.
1Step 1: Understanding the Problem
Determine how many pairs of basketball shorts can be made from 24 yards of nylon if each pair requires \( \frac{3}{4} \) yards of nylon.
2Step 2: Set Up the Division
To find the number of pairs, divide the total amount of nylon (24 yards) by the amount needed per pair (\( \frac{3}{4} \) yards). This can be written as: \[ 24 \div \frac{3}{4} \]
3Step 3: Convert Division to Multiplication
When dividing by a fraction, multiply by its reciprocal. Therefore, the problem becomes: \[ 24 \times \frac{4}{3} \]
4Step 4: Perform the Multiplication
Multiply the numbers: \[ 24 \times \frac{4}{3} = \frac{96}{3} \]
5Step 5: Simplify the Result
Divide the numerator by the denominator: \[ \frac{96}{3} = 32 \]
6Step 6: Interpret the Result
The calculation shows that 32 pairs of shorts can be made from 24 yards of nylon.
Key Concepts
multiplying fractionsreciprocal of a fractionsimplifying fractions
multiplying fractions
Multiplying fractions is a straightforward process. You simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together.
For example: \( \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \).
Always make sure to write your final answer in simplest form if possible.
Remember, fractions can represent parts of a whole, so multiplying them gives us a smaller part of that whole.
For example: \( \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \).
Always make sure to write your final answer in simplest form if possible.
Remember, fractions can represent parts of a whole, so multiplying them gives us a smaller part of that whole.
reciprocal of a fraction
The reciprocal of a fraction is simply switching the numerator and the denominator.
For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
Reciprocal fractions are very useful when it comes to division. Instead of dividing by a fraction, you can multiply by its reciprocal.
This simplifies the process, particularly for complex fractions.
Always remember: the product of a fraction and its reciprocal is 1. So, \( \frac{3}{4} \times \frac{4}{3} = 1 \).
For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
Reciprocal fractions are very useful when it comes to division. Instead of dividing by a fraction, you can multiply by its reciprocal.
This simplifies the process, particularly for complex fractions.
Always remember: the product of a fraction and its reciprocal is 1. So, \( \frac{3}{4} \times \frac{4}{3} = 1 \).
simplifying fractions
Simplifying fractions means to reduce them to their simplest form where the numerator and denominator are as small as possible.
This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For example: \( \frac{96}{3} = 32 \), because both 96 and 3 can be divided by 3.
Another example: \( \frac{10}{20} \) can be simplified to \( \frac{1}{2} \) by dividing both the numerator and denominator by 10 (their GCD).
Simplifying fractions helps in making the calculations more manageable and easier to understand.
This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For example: \( \frac{96}{3} = 32 \), because both 96 and 3 can be divided by 3.
Another example: \( \frac{10}{20} \) can be simplified to \( \frac{1}{2} \) by dividing both the numerator and denominator by 10 (their GCD).
Simplifying fractions helps in making the calculations more manageable and easier to understand.
Other exercises in this chapter
Problem 43
Write exponential notation: \(4 \cdot 4 \cdot 4 \cdot 4 \cdot 4\)
View solution Problem 43
Solve. $$ 3008=x+2134 $$
View solution Problem 44
Determine whether each number is prime, composite, or neither. $$ 49 $$
View solution Problem 44
Simplify. $$ \frac{8}{8} $$
View solution