Dividing a Decimal by a Whole Number

Calculate $12.48 \div 4$

Set up the division: Place $12.48$ under the division symbol with $4$ outside = $4 \overline{)12.48}$

Divide the whole number part: $12 \div 4 = 3$ = Write $3$ above the $2$

Place the decimal point: Put the decimal directly above where it appears in $12.48$ = $3.$

Divide the tenths: $4 \div 4 = 1$ = $3.1$

Divide the hundredths: $8 \div 4 = 2$ = $3.12$

Answer: $12.48 \div 4 = 3.12$

Dividing by a Decimal (Making the Divisor Whole)

Calculate $8.4 \div 0.4$

Identify the problem: We cannot easily divide by $0.4$, so we need to make it a whole number = Make divisor whole

Multiply both numbers by 10: $8.4 \times 10 = 84$ and $0.4 \times 10 = 4$ = $\frac{84}{4}$

Verify the fraction is equivalent: $\frac{8.4}{0.4} = \frac{84}{4}$ (multiplying top and bottom by 10 keeps the value the same) = Equivalent division

Divide the whole numbers: $84 \div 4 = 21$ = $21$

Answer: $8.4 \div 0.4 = 21$

Dividing with Two Decimal Places in the Divisor

Calculate $15.75 \div 0.25$

Count decimal places in divisor: $0.25$ has 2 decimal places = Multiply by $100$

Multiply both numbers by 100: $15.75 \times 100 = 1575$ and $0.25 \times 100 = 25$ = $\frac{1575}{25}$

Divide: $1575 \div 25 = 63$ = $63$

Verify with estimation: $16 \div 0.25 = 64$, so $63$ is reasonable = Answer checks out

Answer: $15.75 \div 0.25 = 63$

Division Resulting in a Decimal Quotient

Calculate $7 \div 4$

Set up the division: $4 \overline{)7.00}$ (add decimal and zeros) = Ready to divide

Divide 7 by 4: $7 \div 4 = 1$ remainder $3$ = $1.$

Bring down the zero, divide 30 by 4: $30 \div 4 = 7$ remainder $2$ = $1.7$

Bring down another zero, divide 20 by 4: $20 \div 4 = 5$ remainder $0$ = $1.75$

Answer: $7 \div 4 = 1.75$

Mistake: Forgetting to move the decimal point in the dividend when making the divisor whole

Why: When you multiply the divisor by 10 (or 100), you must do the same to the dividend to keep the division equivalent.

Correct: $8.4 \div 0.4$: Multiply BOTH by 10 to get $84 \div 4 = 21$

Mistake: Placing the decimal point in the wrong position in the quotient

Why: The decimal in the quotient must align directly above where it appears in the dividend.

Correct: In $12.48 \div 4$, place the decimal after the $3$: $3.12$, not $31.2$ or $0.312$

Mistake: Stopping the division too early when there is a remainder

Why: If there is a remainder, add zeros after the decimal and continue dividing.

Correct: $7 \div 4$: Don't stop at $1$ R$3$. Add zeros: $7.00 \div 4 = 1.75$

Mistake: Thinking dividing by a decimal less than 1 makes the answer smaller

Why: Dividing by a number less than 1 actually makes the answer larger. Think: how many $0.5$s fit into $3$? Six!

Correct: $3 \div 0.5 = 6$ (larger than 3, not smaller)

Unit Price Calculation

Stores use decimal division to calculate the price per item or per unit weight.

A pack of 6 yogurts costs 4.50 euros. Each yogurt costs $4.50 \div 6 = 0.75$ euros.

Splitting Costs

Friends often need to divide costs equally, including amounts with decimals.

A dinner bill of 87.50 euros split 5 ways: $87.50 \div 5 = 17.50$ euros each.

Q: Why do we multiply both numbers by 10 when dividing by a decimal?

A: Multiplying the numerator and denominator of a fraction by the same number keeps the value the same. $\frac{8.4}{0.4} = \frac{84}{4}$ because both top and bottom were multiplied by 10.

Q: How do I know when to stop dividing?

A: Stop when: (1) there is no remainder, (2) you reach a repeating pattern, or (3) you have enough decimal places for your purpose (often 2 for money).

Q: Why is $3 \div 0.5 = 6$ instead of a smaller number?

A: Think of it as: 'How many halves fit into 3?' Six halves make 3 wholes. Dividing by a fraction or decimal less than 1 asks how many small pieces fit into the whole.