Dividing whole numbers, Section 2 -- A complete course in arithmetic (2)

S k i l l
i n
A R I T H M E T I C

SHORT DIVISION

Lesson 11 Section 2

4.	How do we round off, or approximate, a decimal to a given number of decimal places?

$6.738$6.74

	Look at the digit to the right of the given place. If it is a 5 or greater, add 1 to the given place. If it is less than 5, leave the given place unchanged. In either case, drop all the remaining digits.

Example 1. Round off this decimal 7.253896 to three decimal places.

Answer. 7.2538967.254

(The wavy equal sign means "is approximately equal to.")

To round off to three decimal places, we must look at the digit in the fourth place. The digit in the fourth place is 8 (greater than 5). Therefore, we add 1 to the previous digit 3.

Example 2. Round off 7.253896 to two decimal places.

Answer. 7.2538967.25

The digit in the third decimal place is 3 (less than 5). Therefore, we leave the digit in the second place (5) unchanged.

Example 3. Round off 7.253896 to the nearest tenth.

Answer. 7.2538967.3

To round off to the nearest tenth, means to keep one decimal place. (To round off to the nearest hundredth would mean to keep two places; to the nearest thousandth, three; and so on. Lesson 2, Question 6.)

Now, the digit in the second decimal place is 5. Therefore add 1 to the previous digit 2.

Example 4. Round off $6.497014

Answer. This is money. Therefore we must round off to two decimal places:

$6.497014$6.50

The digit in the third decimal place is 7 (greater than 5). Therefore, when we add 1 to 9 of 6.49, we get 6.50

To round off whole numbers, see Lesson 1.

5.	How do we express the quotient as a decimal?


	When there are no more digits in the dividend:
	1)	Place a decimal point and bring it up to the quotient.
	2)	Add a 0 onto the remainder.
	3)	Continue dividing. Add a 0 onto each remainder until the division is exact. If the division will never be exact, round off the quotient to the required number of decimal places.

Begin,

"4 goes into 32 eight (8) times.

"4 goes into 7 one (1) time with 3 left over.

There are no more digits in the dividend. Therefore, according to the rule: Place a decimal point, bring it up to the quotient, and add a 0 onto the remainder 3:

Continue dividing. Place a 0 after each remainder:

The quotient of 327 ÷ 4, then, may be expressed in three different ways:

Example 6. Compare the following:

a)	If it takes 4 yards of material to cover a sofa, how many sofas could you cover with 25 yards?
b)	If 4 people go to lunch and the bill is $25, how much does each one pay?

Solution.

a) We must divide by 4:

"4 goes into 25 six (6) times with 1 left over."

This means that you could cover 6 sofas, and 1 yard of material will remain.

b)	Again, we must divide by 4, but here we must express the quotient as a decimal – as money – and 6 R 1 makes no sense.

Each person will pay $6.25.

Solution. "6 goes into 12 two (2) times. 6 goes into 7 one (1) time with 1 left over." Place a decimal point, and add a 0 onto the remainder 1:

Now the same remainder 4 will be repeated. This division will never be exact. To round off to three decimal places, then, we must know the digit in the fourth place -- 6. Therefore, we add 1 to the previous digit; we get 21.167

To round off, we must know one more decimal place
than is required.

Example 8. The dividend less than the divisor.

5 ÷ 8

The dividend 5 is less than the divisor 8. In this case, proceed in this same way. 8 is the divisor. Write 8 outside the division box:

"8 goes into 5 zero (0)." Place the decimal point, and add on a 0.

Next, "8 goes into 50 six (6) times (48) with 2 left over."

"8 goes into 20 two (2) times (16) with 4 left over."

"8 goes into 40 five (5) times exactly."

We will see the actual application of the dividend less than the divisor when we come to changing a fraction to a decimal (Lesson 23). For,

5
8

5 ÷ 8.

The horizontal line separating 5 and 8 is the division bar.

Example 9. The following division will not be exact. Round it off to three decimal places:

1 ÷ 11

Solution. Place 11 outside the box and begin:

"11 goes into 1 zero (0)." Place the decimal point, and add on a 0.

Now 11 does not go into 10. Write 0 in the quotient --

-- and add another 0 onto the dividend.

"11 goes into 100 nine (9) times, with 1 left over."

Write 9 in the quotient, and add a 0 onto the remainder 1.

Write 0 in the quotient, and add another 0 to the dividend. Again,

This division will never be exact. We will keep getting 090909. To round it off to three places, we have calculated the digit in the fourth place. It's 9. Therefore, we will add 1 to the previous digit:

0.09090.091

One sometime sees 1 ÷ 11 = 0.090909... The three dots, called ellipsis, mean "and so on for as far as you please according to the indicated pattern, or according to the rule."

This means that we cannot express 1 ÷ 11 exactly as a decimal. However, we can approximate it to as many decimal places as we please by following the pattern 090909.

Calculator problems

Example 10. Patricia pays $66 each year for magazine subscriptions. How much does that cost her each week?

Solution. We must divide $66 by the number of weeks in a year, 52. Press

See

1.2692307

Since this is money, we must round it off to the nearest cent.

1.2692307$1.27

It costs Patricia approximately $1.27 each week.

Example 11. A plane flew between New York and Los Angeles, a distance of 2,477 miles, in 4.25 hours. What was its speed in miles per hour?

Solution. Miles "per" hour indicates Miles ÷ hours (Lesson 10). Press