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

When we multiply decimals, the total decimal places in the product matter.

Step 1 — Decimal Places in Product

We look at the digits after the decimal point. We count these digits for the multiplier and the multiplicand. Then we count them for the product. For $9.5 \times 5$ : Multiplier (9.5) has 1 decimal digit. Multiplicand (5) has 0 decimal digits. Product (47.5) has 1 decimal digit. We see that $1 + 0 = 1$ .

For $12.5 \times 7.5$ : Multiplier (12.5) has 1 decimal digit. Multiplicand (7.5) has 1 decimal digit. Product (93.75) has 2 decimal digits. We see that $1 + 1 = 2$ .

For $1.64 \times 6$ : Multiplier (1.64) has 2 decimal digits. Multiplicand (6) has 0 decimal digits. Product (9.84) has 2 decimal digits. We see that $2 + 0 = 2$ .

For $5.7 \times 13.35$ : Multiplier (5.7) has 1 decimal digit. Multiplicand (13.35) has 2 decimal digits. Product (76.095) has 3 decimal digits. We see that $1 + 2 = 3$ .

$\boxed{\text{The product has a total of (multiplier + multiplicand) decimal digits.}}$

Step 2 — Zeroes in the Denominator

Let us look at the number of zeroes in the denominator. We convert the decimals to fractions. For $9.5 \times 5 = \frac{95}{10} \times 5 = \frac{475}{10}$ : The denominator is 10. It has 1 zero. This matches the 1 decimal digit in the product.

For $12.5 \times 7.5 = \frac{125}{10} \times \frac{75}{10} = \frac{9375}{100}$ : The denominator is 100. It has 2 zeroes. This matches the 2 decimal digits in the product.

For $1.64 \times 6 = \frac{164}{100} \times 6 = \frac{984}{100}$ : The denominator is 100. It has 2 zeroes. This matches the 2 decimal digits in the product.

For $5.7 \times 13.35 = \frac{57}{10} \times \frac{1335}{100} = \frac{76095}{1000}$ : The denominator is 1000. It has 3 zeroes. This matches the 3 decimal digits in the product.

$\boxed{\text{The product's denominator zeroes match its decimal digits.}}$

Answer

(i) The product's decimal digits sum the multiplier's and multiplicand's. (ii) The number of zeroes in the product's denominator equals its decimal digits. (iii) The product's denominator zeroes sum the multiplier's and multiplicand's zeroes.

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