Why can you use fraction multiplication to check the answer to a fraction division problem? How do you use multiplication to check a division problem?

Answer 1

When you perform a division with fractions, you can check your answer using multiplication based on the relationship between these operations. Specifically, division and multiplication are inverse operations. This means that if you divide a fraction by another fraction and get a result, you can multiply that result by the divisor to check whether you get back the original dividend.

Why Fraction Multiplication Checks Fraction Division

1. Inverse Operations: When we divide by a fraction, we are essentially finding out how many times that fraction fits into the whole. The multiplication operation reverses this process.

2. Equation Relationship: If you have a division problem like $\frac{a}{b}÷\frac{c}{d}$, you can express it as the equation: $$\frac{a}{b}÷\frac{c}{d}=x$$ This means you are asking, "What is $x$ such that when $x$ is multiplied by $\frac{c}{d}$, it equals $\frac{a}{b}$?" In terms of an equation, you can write: $$x\cdot \frac{c}{d}=\frac{a}{b}$$

3. Checking the Answer: To check your answer $x$:

   • Multiply $x$ by the divisor $\frac{c}{d}$.
   • Confirm whether the result equals the original dividend $\frac{a}{b}$.

How to Use Multiplication to Check a Division Problem

Let's go through an example to see how this works:

Example

Suppose you want to divide $\frac{2}{3}$ by $\frac{4}{5}$ and found the answer to be $\frac{5}{6}$.

1. Division Problem: $$\frac{2}{3}÷\frac{4}{5}=\frac{5}{6}$$

2. Check Your Answer Using Multiplication:

   • Multiply the answer $\frac{5}{6}$ by the divisor $\frac{4}{5}$: $$\frac{5}{6}\cdot \frac{4}{5}$$ When multiplying fractions, multiply the numerators and the denominators: $$\frac{5\cdot 4}{6\cdot 5}=\frac{20}{30}$$ Simplifying $\frac{20}{30}$ gives you $\frac{2}{3}$.

3. Verification:

   • Since multiplying the result $\frac{5}{6}$ by the divisor $\frac{4}{5}$ gives you back $\frac{2}{3}$ (the original dividend), you have confirmed that your division was correct.

Summary

In summary, you can use multiplication to check fraction division problems because multiplication is the inverse of division. By multiplying the quotient you obtained by the divisor, you should arrive back at the original dividend, verifying that your division was done correctly.