We know that two fractions are equivalent when the numerators and denominators are related by the same factor, and when dividing the numerator by the denominator gives the same quotient. For example, we can tell that $\frac{6}{4}$ and $\frac{60}{40}$ are equivalent fractions because:

• Dividing 6 by 4 and dividing 60 by 40 both give 1.5.
• The numerators and denominators of $\frac{6}{4}$ and $\frac{60}{40}$ are related by the same factor of 10: $\frac{6 \boldcdot 10}{4 \boldcdot 10} = \frac{60}{40}$.

Division expressions can also be equivalent. For example, the expression $5,400 \div 900$ is equivalent to $54 \div 9$ because:

• They both have a quotient of 6.
• The dividends and divisors in $5,400 \div 900$ and $54 \div 9$ are related by the same factor of 100: $54 \boldcdot 100 = 5,400$ and $9 \boldcdot 100 = 900$.

This means that an expression such as $5.4 \div 0.9$ also has the same value as $54 \div 9$. The dividend and divisor in $54 \div 9$ are each 10 times those in $5.4 \div 0.9$, but their quotients are the same.

This understanding can help us divide a decimal dividend by a decimal divisor: We can multiply each decimal by the same power of 10 so that both the dividend and the divisor are whole numbers, and then we divide the whole numbers.

For example, to calculate $7.65 \div 1.2$ we can multiply each decimal by 100, and then calculate $765 \div 120$. Here is the calculation with long division:

Long division of 765 divided by 120. 10 rows. Top row: 6 point 3 7 5. Second row: 120, long division symbol with 765 inside. Third row: minus 720. Horizontal line. Fourth row: 45 point 0. Fifth row: minus 36 point 0. Horizontal line. Sixth row: 9 point 0 0. Seventh row: minus 8 point 4 0. Horizontal line. Eighth row: point 6 0 0. Ninth row: minus point 6 0 0. Horizontal line. Tenth row: 0.

Because the expression $765 \div 120$ is equivalent to $7.65 \div 1.2$, we know that 6.375 is also a quotient of $7.65 \div 1.2$.