Another Peek Beyond the Point · Lesson 10 of 13
Division with a Decimal Divisor
“Make a decimal divisor a whole number while preserving the quotient, then divide and check.”
• Explain why multiplying both dividend and divisor by the same non-zero number preserves their quotient. • Choose a power of ten that makes a decimal divisor a whole number. • Divide decimal quantities in speed and equal-group situations. • Check the scale and units of a division answer.
Make the divisor easier to use
Sharing into 1.3 groups is harder to picture than sharing into a whole number of groups. A measurement question can still lead naturally to division by 1.3: for example, asking how many lengths of 1.3 m fit into 4.68 m. We can describe both lengths in decimetres instead. Because 1 m = 10 decimetres, the same question becomes 46.8 ÷ 13. Changing both measurements to the same smaller unit changes the written numbers but preserves the number of groups.
The same idea works without units. Consider 12 ÷ 3 = 4. Doubling both numbers gives 24 ÷ 6 = 4; making both ten times as large gives 120 ÷ 30 = 4. Each group and the total grow together, so the number of groups stays unchanged. This is the equivalent-fraction rule applied to division: 12/3 and 120/30 have the same value.
Problem
Find 4.68 ÷ 1.3.
- 1.The divisor 1.3 has one decimal place. Multiplying it by 10 gives the whole number 13.
- 2.Multiply the dividend by the same 10: 4.68 × 10 = 46.8. Therefore 4.68 ÷ 1.3 = 46.8 ÷ 13.
- 3.13 fits into 46 three times, leaving 7 ones. Combine those with the 8 tenths to get 78 tenths.
- 4.78 tenths ÷ 13 = 6 tenths. The quotient is 3.6.
- 5.Check with the original numbers: 1.3 × 3.6 = 4.68.
Choose the scale factor carefully
Count decimal places in the divisor, not in the dividend. A divisor in hundredths becomes a whole number after multiplication by 100. Apply that same multiplication to the dividend even if it has a different number of decimal places. You may then have a whole-number dividend or a decimal dividend; both are handled by the place-value division you already know.
Problem
Compare 4.68 ÷ 1.3 and 4.68 ÷ 0.13.
- 1.The first quotient is 3.6, as shown above.
- 2.For 0.13, use a factor of 100: 4.68 × 100 = 468 and 0.13 × 100 = 13.
- 3.Divide 468 by 13. Since 13 × 36 = 468, the second quotient is 36.
- 4.The divisor 0.13 is one-tenth of 1.3, so ten times as many of these smaller groups fit into 4.68. This explains why the quotient becomes ten times as large.
| Original division | Scale BOTH numbers by | Equivalent division | Quotient |
|---|---|---|---|
| 2.46 ÷ 1.5 | 10 | 24.6 ÷ 15 | 1.64 |
| 2.46 ÷ 0.15 | 100 | 246 ÷ 15 | 16.4 |
| 2.46 ÷ 0.015 | 1000 | 2460 ÷ 15 | 164 |
An alternative fraction explanation leads to the same result. For 4.68 ÷ 1.3, write 468/100 divided by 13/10. Dividing by 13/10 means multiplying by its reciprocal 10/13; the reciprocal is the fraction obtained by interchanging its numerator and denominator. The result is (468 × 10)/(100 × 13) = 468/130 = 46.8/13. The useful school method is therefore justified by fraction arithmetic as well as by changing measurement units.
Scaling only one number changes the quotient. Changing 4.68 ÷ 1.3 into 468 ÷ 13 is incorrect: the dividend was multiplied by 100 but the divisor by only 10. Match the scale factors, not the final positions of the decimal points.
Use decimal division in a rate problem
A rate describes an amount for one unit of another quantity. Average speed is total distance divided by total time. If a trip lasts a decimal number of hours, the time is still the divisor. Keep the original units in your final answer even though you temporarily scale the numerical division.
Problem
A train travels 126 km in 2.5 hours. Find its average speed.
- 1.Average speed = distance ÷ time, so calculate 126 ÷ 2.5.
- 2.Multiply both numbers by 10: 1260 ÷ 25.
- 3.25 × 50 = 1250, leaving 10 ones. Regroup 10 ones as 100 tenths; 100 ÷ 25 = 4 tenths.
- 4.The quotient is 50.4. The average speed is 50.4 km per hour, written 50.4 km/h.
- 5.Check: 50.4 × 2.5 = 126. The distance is restored.
Quiz
Which division has the same quotient as 5.728 ÷ 1.52?
What is 4.68 ÷ 0.13?
How should 24.86 ÷ 1.2 be rewritten to make its divisor a whole number?
Why does scaling both numbers equally preserve a quotient?
What is the average speed for 126 km in 2.5 hours?
Practice Problems
- Find 4.68 ÷ 1.3 and 4.68 ÷ 0.13, then explain their relationship.
- Find 2.46 ÷ 1.5, 2.46 ÷ 0.15, and 2.46 ÷ 0.015. Check each answer.
- Explain why 24.6 ÷ 1.5 and 2.46 ÷ 0.15 are equal.
- Rewrite 5.728 ÷ 1.52 using a whole-number divisor. Carry out the first six decimal places without assuming the division will end.
- Find the first five decimal places of 24.86 ÷ 1.2. Keep track of the remainders.
- A vehicle travels 234.45 km using 12.6 litres of fuel. Express its fuel efficiency as an exact quotient in km/L, then calculate four decimal places.
- A 2.75 m ribbon is cut into lengths of 0.25 m. How many lengths are obtained? Explain which division represents the question.
Key Takeaways
• A quotient stays unchanged when both numbers are multiplied by the same non-zero factor. • Choose a power of ten from the decimal places in the divisor. • After making the divisor whole, continue ordinary place-value division. • Scaling just one number changes the answer. • Check the quotient by multiplying it by the original divisor, and include the appropriate units.