29.1: Using the Partial Quotients Method

Lesson

Let's divide whole numbers.

Exercise \(\PageIndex{1}\): Using Base-Ten Diagrams to Calculate Quotients

Elena used base-ten diagrams to find \(372\div 3\). She started by representing \(372\).

Figure \(\PageIndex{1}\)

She made 3 groups, each with 1 hundred. Then, she put the tens and ones in each of the 3 groups. Here is her diagram for \(372\div 3\).

Figure \(\PageIndex{2}\)

Discuss with a partner:

• Elena’s diagram for 372 has 7 tens. The one for \(372\div 3\) has only 6 tens. Why?
• Where did the extra ones (small squares) come from?

Exercise \(\PageIndex{2}\): Using the Partial Quotients Method to Calculate Quotients

1. Andre calculated \(657\div 3\) using a method that was different from Elena’s.

Figure \(\PageIndex{3}\): Method of calculating 657 divided by 3, 4 steps. First step, 1 row, 3, long division symbol with 657 inside. Second step, 4 rows. First row: 200. Second row. 3, long division symbol with 657 inside. Third row: minus 600. Horizontal line. Fourth row: 57. Third step, 10 rows. First row: 9. Second row: 10. Third row: 200. Fourth row: 3, long division symbol with 657 inside. Fifth row: minus 600. Horizontal line. Sixth row: 57. Seventh row: minus 30. Horizontal line. Eighth row: 27. Ninth row: minus 27. Horizontal line. Tenth row: 0. Fourth step: Same as third step, but with 219 at the top row.

1. Andre subtracted 600 from 657. What does the 600 represent?
2. Andre wrote 10 above the 200, and then subtracted 30 from 57. How is the 30 related to the 10?
3. What do the numbers 200, 10, and 9 represent?
4. What is the meaning of the 0 at the bottom of Andre’s work?

2. How might Andre calculate \(896\div 4\)? Explain or show your reasoning.

Exercise \(\PageIndex{3}\): What's the Quotient?

1. Find the quotient of \(1,332\div 9\) using one of the methods you have seen so far. Show your reasoning.
2. Find each quotient and show your reasoning. Use the partial quotients method at least once.
   1. \(1,115\div 5\)
   2. \(665\div 7\)
   3. \(432\div 16\)

Summary

We can find the quotient \(345\div 3\) in different ways.

One way is to use a base-ten diagram to represent the hundreds, tens, and ones and to create equal-sized groups.

Figure \(\PageIndex{4}\)

We can think of the division by 3 as splitting up 345 into 3 equal groups.

Figure \(\PageIndex{5}\)

Each group has 1 hundred, 1 ten, and 5 ones, so \(345\div 3=115\). Notice that in order to split 345 into 3 equal groups, one of the tens had to be unbundled or decomposed into 10 ones.

Another way to divide 345 by 3 is by using the partial quotients method, in which we keep subtracting 3 groups of some amount from 345.

Figure \(\PageIndex{6}\): 2 partial quotients methods of 345 divided by 3. First calculation, 11 rows. First row: 115. Second row: 5. Third row: 10. Fourth row: 100. Fifth row: 3, long division symbol with 345 inside. Sixth row: minus 300, 3 groups of 100. Horizontal line. Seventh row: 45. Eighth row: minus 30, 3 groups of 10. Horizontal line. Ninth row: 15. Tenth row: minus 15, 3 groups of 5. Horizontal line. Eleventh row: 0. Second calculation, 11 rows. First row: 115. Second row: 50. Third row: 50. Fourth row: 15. Fifth row: 3, long division symbol with 345 inside. Sixth row: minus 45, 3 groups of 15. Horizontal line. Seventh row: 300. Eighth row: minus 150, 3 groups of 50. Horizontal line. Ninth row: 150. Tenth row: minus 150, 3 groups of 50. Horizontal line. Eleventh row: 0

• In the calculation on the left, first we subtract 3 groups of 100, then 3 groups of 10, and then 3 groups of 5. Adding up the partial quotients (\(100+10+5\)) gives us 115.
• The calculation on the right shows a different amount per group subtracted each time (3 groups of 15, 3 groups of 50, and 3 more groups of 50), but the total amount in each of the 3 groups is still 115. There are other ways of calculating \(345\div 3\) using the partial quotients method.

Both the base-ten diagrams and partial quotients methods are effective. If, however, the dividend and divisor are large, as in \(1,248\div 26\), then the base-ten diagrams will be time-consuming.

Practice

Exercise \(\PageIndex{4}\)

Here is one way to find \(2,105\div 5\) using partial quotients. Show a different way of using partial quotients to divide 2,105 by 5.

Figure \(\PageIndex{7}\): Partial quotient method of 2,105 divided by 5, 11 rows. First row: 421. Second row: 1. Third row: 20. Fourth row: 400. Fifth row: 5, long division symbol with 2,105 inside. Sixth row: minus 2,000. Horizontal line. Seventh row: 105. Eighth row: minus 100. Horizontal line. Ninth row: 5. Tenth row: minus 5. Horizontal line. Eleventh row: 0.

Exercise \(\PageIndex{5}\)

Andre and Jada both found \(657\div 3\) using the partial quotients method, but they did the calculations differently, as shown here.

Figure \(\PageIndex{8}\): Andre and Jada's calculation of a partial product problem, 657 divided by 3. Andre's calculation. 657 inside the division bar, divided by 3. 3 divides into 657, 200 times. Subtract 600 from 657 for a remainder of 57. 3 divides into 57, 10 times. Subtract 30 from 57 for a remainder of 27. 3 divides into 27, 9 times. Subtract 27 from 27 for a remainder of 0. Add all of Andre's partial quotients of 200, 10 and 9 for a total of 219. Jada's calculation. 657 inside the division bar, divided by 3. 3 divides into 657, 50 times. Subtract 150 from 657 for a remainder of 507. 3 divides into 507, 100 times. Subtract 300 from 507 for a remainder of 207. 3 divides into 207, 60 times. Subtract 180 from 207 for a remainder of 27. 3 divides into 27, 9 times. Subtract 27 from 27 for a remainder of 0. Add all of Jada's partial quotients of 50, 100, 60 and 9 for a total of 219.

1. How is Jada's work the same as Andre’s work? How is it different?
2. Explain why they have the same answer.

Exercise \(\PageIndex{6}\)

Which might be a better way to evaluate \(1,150\div 46\): drawing base-ten diagrams or using the partial quotients method? Explain your reasoning.

Exercise \(\PageIndex{7}\)

Here is an incomplete calculation of \(534\div 6\).

Write the missing numbers (marked with “?”) that would make the calculation complete.

Figure \(\PageIndex{9}\): An incomplete calculation of 534 divided by 6, 8 rows. First row: 89. Second row: 9. Third row: 80. Fourth row: 6, long division symbol with 534 inside. Fifth row: minus question mark symbol. Horizontal line. Sixth row: question mark symbol. Seventh row: minus question mark symbol. Horizontal line. Eighth row: 0.

Exercise \(\PageIndex{8}\)

Use the partial quotients method to find \(1,032\div 43\).

Exercise \(\PageIndex{9}\)

Which of the polygons has the greatest area?

1. A rectangle that is 3.25 inches wide and 6.1 inches long.
2. A square with side length of 4.6 inches.
3. A parallelogram with a base of 5.875 inches and a height of 3.5 inches.
4. A triangle with a base of 7.18 inches and a height of 5.4 inches.

(From Unit 5.3.4)

Exercise \(\PageIndex{10}\)

One micrometer is a millionth of a meter. A certain spider web is 4 micrometers thick. A fiber in a shirt is 1 hundred-thousandth of a meter thick.

1. Which is wider, the spider web or the fiber? Explain your reasoning.
2. How many meters wider?

(From Unit 5.2.3)