Another Peek Beyond the Point · Lesson 6 of 13
From Fractions to Decimal Quotients
“Express division as a fraction and find decimal answers by choosing equivalent denominators.”
• Represent a whole-number division as a fraction. • Create equivalent fractions with denominators 10, 100, or 1000. • Separate a whole part from a remaining fractional part when sharing. • Recognise why a more general division method is sometimes needed.
A quotient can be written as a fraction
Dividing 29 m of ribbon between two people gives 29 ÷ 2 metres each. The fraction 29/2 expresses exactly the same division. A fraction can be larger than one: here it includes a whole-number part and a half metre. We can find the decimal by sharing first or by finding an equivalent fraction.
The result of a division. In 29 ÷ 2 = 14.5, the quotient is 14.5.
If each person receives 14 m, 28 m have been used and 1 m remains. Half of that metre is 0.5 m, because 1/2 = 5/10. The remainder is still part of the ribbon and must be shared. It is not an extra whole metre added to each person’s share.
Problem
Find the decimal length each person receives.
- 1.Give each person 14 m. This uses 14 × 2 = 28 m and leaves 1 m.
- 2.Each receives another 1/2 m = 5/10 m = 0.5 m.
- 3.Each share is 14 + 0.5 = 14.5 m.
- 4.Equivalently, 29/2 = (29 × 5)/(2 × 5) = 145/10 = 14.5. Check: 14.5 × 2 = 29.
Choose an equivalent denominator
A fraction becomes easy to write as a decimal when its denominator is 10, 100, or 1000. Multiply its numerator and denominator by the same number to keep the value unchanged. The multiplier must make the denominator a suitable power of ten. For a denominator of 4, multiplying by 25 makes 100; multiplying by 5 would make 20, which is less convenient for direct decimal writing.
Problem
Find 29 ÷ 4 as a decimal.
- 1.Represent the division as 29/4.
- 2.Because 4 × 25 = 100, multiply the numerator and denominator by 25.
- 3.(29 × 25)/(4 × 25) = 725/100 = 7.25.
- 4.Each person receives 7.25 m. Four shares give 7.25 × 4 = 29 m.
| Original fraction | Useful common multiplier | Equivalent decimal fraction | Decimal |
|---|---|---|---|
| 1/2 | 5 | 5/10 | 0.5 |
| 29/4 | 25 | 725/100 | 7.25 |
| 2/5 | 2 | 4/10 | 0.4 |
| 5/8 | 125 | 625/1000 | 0.625 |
Problem
Express 5/8 as a decimal.
- 1.8 is not a factor of 10 or 100, but 8 × 125 = 1000.
- 2.Multiply both parts of the fraction by 125: 5/8 = 625/1000.
- 3.625 thousandths are 0.625. As a check, 0.625 × 8 = 5.
Why this method has limits
The equivalent-fraction method is efficient when a suitable denominator is easy to obtain. But it is not always possible to multiply the denominator by a whole number to make 10, 100, or 1000. For example, none of these powers of ten is divisible by 3. So 10/3 cannot be converted to a decimal fraction in this way. That does not make the division meaningless; it tells us we need another method.
Long division provides that more general method by sharing each place value and regrouping anything left over. Sometimes it finishes; sometimes the decimal continues. For now, keep the limitation separate from a calculation error. A correct fraction such as 10/3 still represents an exact quantity even if you cannot write it with a short finite decimal.
Problem
A learner changes 29/4 into 29/100 and writes 0.29. What went wrong?
- 1.The denominator was multiplied by 25, but the numerator was not.
- 2.To preserve the fraction, both must be multiplied by 25: 29/4 = 725/100.
- 3.The correct decimal is 7.25. The value 0.29 would give only 1.16 when multiplied by 4, rather than 29.
Equivalent fractions require the same multiplier in numerator and denominator. Making only the denominator look convenient changes the quantity being divided.
Quiz
Which fraction represents 29 ÷ 4?
What multiplier turns a denominator of 4 into 100?
What is 29 ÷ 2?
Which decimal equals 5/8?
Why is 10/3 not directly converted to a fraction with denominator 100?
Practice Problems
- Convert 2/5, 13/4, 4/50, and 5/8 into decimals using equivalent fractions.
- Explain why 29 ÷ 2 can be found by sharing 28 m first and then sharing the remaining metre.
- Find 18/5, 415/4, 1217/2, and 4827/8 as decimals.
- A 3 m ribbon is shared among eight people. Use an equivalent fraction to find one person’s share.
- Explain why changing 7/4 to 7/100 is not a valid simplification.
- Try making a denominator of 3 into 10, 100, or 1000 with a whole-number multiplier. Explain why a different division method is needed.
Key Takeaways
• A fraction and a division can describe the same quotient. • Equivalent fractions preserve value by scaling numerator and denominator equally. • Denominators 10, 100, and 1000 give direct decimal representations. • A remainder from sharing whole units can be shared as fractional units. • Some fractions require long division rather than direct denominator conversion.