Decimals × and ÷ decimals

Japanese school year: Elementary 5

What you learn

You will learn how to carry out multiplication and division where both numbers are decimals. This operation is widely used in calculating areas, unit rates, and scaling values proportionally. Familiarity with multiplying and dividing decimals by whole numbers provides a strong foundation for this topic.

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Key points

This is the procedure for multiplying decimals. Multiply them as whole numbers, then place the decimal point in the product by counting from the right by the total number of decimal places in both factors.

This formula shows how to simplify division by a decimal. Shift the decimal point of divisor $b$ to the right to make it a whole number (multiplying by $10^n$ ), and shift dividend $a$ by the same number of places.

a \div b = (a \times 10^n) \div (b \times 10^n)

This formula finds remainder $r$ in decimal division. When computing an integer quotient $q$ , place the decimal point of remainder $r$ straight down from dividend $a$ 's original position before shifting.

a = b \times q + r

This shows how multiplying or dividing by a positive decimal $b$ between $0$ and $1$ changes the value. Multiplying positive number $a$ by $b$ makes it smaller than $a$ , while dividing by $b$ makes it larger than $a$ .

0 < b < 1 \implies a \times b < a, \quad a \div b > a \quad (a > 0)

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Practice 1Coming soon