Dividing decimal by decimal

The trick: turn the divisor into a whole number by multiplying both numerator and denominator by the same power of ten.

Idea

To compute $a \div b$ where $b$ has decimals, multiply both by $10^k$, where $k$ is the number of decimal places of $b$. This shifts both decimal points right by $k$ and makes $b$ a whole number. The quotient is unchanged.

Example: $4.5 \div 0.3$. Divisor has $1$ decimal place. Multiply both by $10$: $45 \div 3 = 15$. So $4.5 \div 0.3 = 15$.

Example: $0.045 \div 0.5$. Multiply both by $10$: $0.45 \div 5 = 0.09$. So $0.045 \div 0.5 = 0.09$.

Always estimate the quotient first. $4.5 \div 0.3$ , roughly $5 \div 0.3$, which is about $15$. So our answer $15$ is reasonable.

Two equivalent approaches:

1. Multiply both sides by power of ten.
2. Rewrite the division as a fraction $\tfrac{a}{b}$; multiply numerator and denominator by same power of ten.

These give the same answer.

Worked examples

Example 1. Compute $1.2 \div 0.3$.

Multiply both by $10$: $12 \div 3 = 4$. So $1.2 \div 0.3 = 4$.

Example 2. Compute $7.5 \div 0.25$.

Multiply both by $100$: $750 \div 25 = 30$. So $7.5 \div 0.25 = 30$.

Example 3. Compute $0.84 \div 0.4$.

Multiply both by $10$: $8.4 \div 4 = 2.1$. So $0.84 \div 0.4 = 2.1$.

Example 4. Compute $0.1 \div 0.04$.

Multiply both by $100$: $10 \div 4 = 2.5$. So $0.1 \div 0.04 = 2.5$.

Try it yourself

1. Compute $2.4 \div 0.2$.
2. Compute $6.3 \div 0.7$.
3. Compute $0.36 \div 0.06$.
4. Compute $1 \div 0.5$.
5. Compute $0.875 \div 0.05$.
6. Compute $3.6 \div 1.2$.
7. Compute $9.9 \div 0.11$.
8. A piece of cloth $7.5$ m long is cut into pieces of $0.5$ m. How many pieces?

Activity

Use a $1$ m ruler. Cut paper strips of length $0.2$ m. Count how many fit in $1$ m, in $2$ m, in $2.4$ m. Verify by computing $1 \div 0.2$, $2 \div 0.2$, $2.4 \div 0.2$.