Another Peek Beyond the Point · Lesson 8 of 13
Continuing Division Beyond the Ones Place
“Regroup remainders as tenths, hundredths, and thousandths to obtain decimal quotients.”
• Continue long division when a whole-unit remainder is left. • Explain why the quotient point comes before the tenths digit. • Use appended decimal zeroes without changing the dividend. • Verify decimal quotients through multiplication or equivalent fractions.
A leftover whole unit can still be shared
Dividing 1325 by 4 follows the same whole-number steps as dividing 1324 by 4, but one whole unit remains. An answer of 331 with remainder 1 is correct when reporting whole-number groups. If we want the exact size of each share, we can divide that remaining unit too. It is shared among all four parts rather than added whole to each part.
Regroup the one as ten tenths. Each part receives two tenths, using eight tenths and leaving two tenths. Regroup those two tenths as twenty hundredths. Each part then receives five hundredths, and nothing remains. Thus each share is 331 + 0.2 + 0.05 = 331.25.
Problem
Find the quotient as a decimal.
- 1.The whole-number stages give 331, with 1 one left over.
- 2.1 one = 10 tenths. 10 tenths ÷ 4 gives 2 tenths per share and 2 tenths left.
- 3.2 tenths = 20 hundredths. 20 hundredths ÷ 4 gives 5 hundredths per share.
- 4.The quotient is 331.25. The point separates the ones digit 1 from the tenths digit 2.
- 5.Check by fractions: 1325/4 = (1325 × 25)/100 = 33125/100 = 331.25.
Appending zeroes preserves the starting quantity
Writing 1325 as 1325.00 does not add any amount. It shows that there are initially zero tenths and zero hundredths alongside the original number. When a remainder is regrouped, those places become available to record smaller-unit shares. This differs from appending a zero to make 13250, which would multiply the whole quantity by ten.
The decimal point enters the quotient at the boundary between ones and tenths, regardless of how many tenths can be given to each part. If the first decimal quotient digit is zero, write it. The next non-zero digit belongs to a smaller place and must not slide into the tenths place.
Problem
Find 237 ÷ 8.
- 1.Regroup 2 hundreds with 3 tens to make 23 tens. Each part gets 2 tens and 7 tens remain.
- 2.Regroup the 7 tens and add the 7 ones: 77 ones. Each part gets 9 ones and 5 ones remain.
- 3.5 ones become 50 tenths. Each part gets 6 tenths and 2 tenths remain.
- 4.2 tenths become 20 hundredths. Each part gets 2 hundredths and 4 hundredths remain.
- 5.4 hundredths become 40 thousandths. Each part gets 5 thousandths and the remainder is zero.
- 6.The quotient is 29.625. Check: 29.625 × 8 = 237.
| Current amount in 237 ÷ 8 | Digit placed in quotient | Remainder before regrouping |
|---|---|---|
| 23 tens | 2 tens | 7 tens |
| 77 ones | 9 ones | 5 ones |
| 50 tenths | 6 tenths | 2 tenths |
| 20 hundredths | 2 hundredths | 4 hundredths |
| 40 thousandths | 5 thousandths | 0 |
Finish only when the remainder is zero
An exact terminating decimal ends when the division leaves zero remainder. There is then nothing more to share. It does not necessarily end after the same number of places as the dividend originally displayed. A whole-number dividend can give several decimal places, as 237 ÷ 8 demonstrates. Some divisions never reach remainder zero; a later lesson will investigate them.
A decimal representation that finishes after a finite number of places because the long division reaches remainder zero.
Problem
Find 18 ÷ 5.
- 1.18 ones ÷ 5 gives 3 ones per part, with 3 ones remaining.
- 2.Regroup 3 ones as 30 tenths. 30 tenths ÷ 5 gives 6 tenths with no remainder.
- 3.So 18 ÷ 5 = 3.6. Check: 3.6 × 5 = 18.
- 4.The equivalent-fraction method agrees: 18/5 = 36/10 = 3.6.
A remainder of 1 in 1325 ÷ 4 does not make the answer 331.1. It represents one whole unit still shared among four parts, giving another 0.25 to each.
Quiz
What is 1325 ÷ 4?
When does the quotient point belong?
Which value equals 1325?
In 237 ÷ 8, four hundredths are regrouped as what?
Which calculation verifies 29.625 as the quotient of 237 ÷ 8?
Practice Problems
- Find 1325 ÷ 4 and name each decimal digit’s place value.
- Calculate 237 ÷ 8, recording every remainder and regrouping.
- Calculate 1526 ÷ 4 and choose the correct quotient using a magnitude check.
- Calculate 3567 ÷ 8 and verify by multiplication.
- Find 18 ÷ 5 and 415 ÷ 4 using both long division and equivalent fractions.
- Calculate 1217 ÷ 2 and 4827 ÷ 8. Check using equivalent decimal fractions and multiplication.
- Explain why writing 237.000 preserves the dividend but writing 237000 does not.
- Give a division of two whole numbers with a quotient between 0 and 1, and explain the zero before its point.
Key Takeaways
• Whole-number remainders can be shared in smaller decimal units. • One one becomes ten tenths; one tenth becomes ten hundredths. • The quotient point separates ones from tenths. • Appending decimal zeroes preserves the dividend’s value. • A terminating calculation finishes when the remainder becomes zero.