Observe the number of digits after the decimal point in the multiplier, the multiplicand and the product. Also note the number of zeroes in the denominator.

Question

Homi · Accepted Answer

Understand the Question

When multiplying decimal numbers, the total number of decimal places in the product equals the sum of the decimal places in the numbers being multiplied (multiplier and multiplicand).
When decimal numbers are written as fractions, the number of zeroes in the denominator (like $10, 100, 1000$ ) is equal to the number of digits after the decimal point.

Step 1 · Count Decimal Places in Multiplication

For $9.5 \times 5$ :

Multiplier ( $9.5$ ): $1$ decimal digit
Multiplicand ( $5$ ): $0$ decimal digits
Product ( $47.5$ ): $1$ decimal digit $1 + 0 = 1$

For $12.5 \times 7.5$ :

Multiplier ( $12.5$ ): $1$ decimal digit
Multiplicand ( $7.5$ ): $1$ decimal digit
Product ( $93.75$ ): $2$ decimal digits $1 + 1 = 2$

For $1.64 \times 6$ :

Multiplier ( $1.64$ ): $2$ decimal digits
Multiplicand ( $6$ ): $0$ decimal digits
Product ( $9.84$ ): $2$ decimal digits $2 + 0 = 2$

For $5.7 \times 13.35$ :

Multiplier ( $5.7$ ): $1$ decimal digit
Multiplicand ( $13.35$ ): $2$ decimal digits
Product ( $76.095$ ): $3$ decimal digits $1 + 2 = 3$

$\text{Decimal places in product} = (\text{Decimal places in multiplier}) + (\text{Decimal places in multiplicand})$

Step 2 · Count Zeroes in Denominators of Fractional Form

Converting each multiplication into fractional form:

For $9.5 \times 5$ : $\dfrac{95}{10} \times 5 = \dfrac{475}{10}$ Denominator $10$ has $1$ zero, matching the $1$ decimal digit in $47.5$ .

For $12.5 \times 7.5$ : $\dfrac{125}{10} \times \dfrac{75}{10} = \dfrac{9375}{100}$ Denominator $100$ has $2$ zeroes, matching the $2$ decimal digits in $93.75$ .

For $1.64 \times 6$ : $\dfrac{164}{100} \times 6 = \dfrac{984}{100}$ Denominator $100$ has $2$ zeroes, matching the $2$ decimal digits in $9.84$ .

For $5.7 \times 13.35$ : $\dfrac{57}{10} \times \dfrac{1335}{100} = \dfrac{76095}{1000}$ Denominator $1000$ has $3$ zeroes, matching the $3$ decimal digits in $76.095$ .

Answer

The number of decimal places in the product is equal to the sum of the decimal places in the multiplier and multiplicand.

The number of zeroes in the product's denominator equals the total number of decimal digits in the product.

Common Mistakes

Premature Simplification: Canceling or dropping trailing zeroes before placing the decimal point (e.g. $0.50 \times 2 = 1.00$ , which has $2$ decimal places before simplifying to $1$ ).
Confusing Values with Counts: Adding the decimal digits themselves instead of counting the number of places after the decimal point.

Question 7

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