Decimal Operations

Add and Subtract Decimals

Let’s take one more look at the lunch order from the start of Decimals, this time noticing how the numbers were added together.

All three items (sandwich, water, tax) were priced in dollars and cents, so we lined up the dollars under the dollars and the cents under the cents, with the decimal points lined up between them. Then we just added each column, as if we were adding whole numbers. By lining up decimals this way, we can add or subtract the corresponding place values just as we did with whole numbers.

Example

Try it.

Add: \(3.7+12.4.\)

Solution

	\(3.7+12.4\)
Write the numbers vertically so the decimal points line up.	\(\begin{array}{l}3.7 \\ \underset{\text{_____}}{+12.4}\end{array}\)
Place holders are not needed since both numbers have the same number of decimal places.
Add the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.	\(\begin{array}{l}\overset{1}{3}.7 \\ \underset{\text{_____}}{+12.4} \\ 16.1\end{array}\)

Example

Try it.

Add: \(23.5+41.38.\)

Solution

	\(23.5+41.38\)
Write the numbers vertically so the decimal points line up.
Place 0 as a place holder after the 5 in 23.5, so that both numbers have two decimal places.
Add the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.

How much change would you get if you handed the cashier a \(\text{\$20}\) bill for a \(\text{\$14.65}\) purchase? We will show the steps to calculate this in the next example.

Example

Try it.

Subtract: \(20-14.65.\)

Solution

	\(20-14.65\)
Write the numbers vertically so the decimal points line up. Remember 20 is a whole number, so place the decimal point after the 0.
Place two zeros after the decimal point in 20, as place holders so that both numbers have two decimal places.
Subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.

Condensed — the full section is in OpenStax Prealgebra 2e.

Multiply Decimals

Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions.

Do you remember how to multiply fractions? To multiply fractions, you multiply the numerators and then multiply the denominators.

So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side in . Look for a pattern.

	A	B
	\((0.3)(0.7)\)	\((0.2)(0.46)\)
Convert to fractions.	\((\frac{3}{10})(\frac{7}{10})\)	\((\frac{2}{10})(\frac{46}{100})\)
Multiply.	\(\frac{21}{100}\)	\(\frac{92}{1000}\)
Convert back to decimals.	\(0.21\)	\(0.092\)

There is a pattern that we can use. In A, we multiplied two numbers that each had one decimal place, and the product had two decimal places. In B, we multiplied a number with one decimal place by a number with two decimal places, and the product had three decimal places.

How many decimal places would you expect for the product of \((0.01)(0.004)?\) If you said “five”, you recognized the pattern. When we multiply two numbers with decimals, we count all the decimal places in the factors—in this case two plus three—to get the number of decimal places in the product—in this case five.

Once we know how to determine the number of digits after the decimal point, we can multiply decimal numbers without converting them to fractions first. The number of decimal places in the product is the sum of the number of decimal places in the factors.

The rules for multiplying positive and negative numbers apply to decimals, too, of course.

Condensed — the full section is in OpenStax Prealgebra 2e.

Divide Decimals

Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.

To understand decimal division, let’s consider the multiplication problem

\[(0.2)(4)=0.8\]

Remember, a multiplication problem can be rephrased as a division problem. So we can write

\[0.8\div 4=0.2\]

We can think of this as “If we divide 8 tenths into four groups, how many are in each group?” shows that there are four groups of two-tenths in eight-tenths. So \(0.8\div 4=0.2.\)

Using long division notation, we would write

Notice that the decimal point in the quotient is directly above the decimal point in the dividend.

To divide a decimal by a whole number, we place the decimal point in the quotient above the decimal point in the dividend and then divide as usual. Sometimes we need to use extra zeros at the end of the dividend to keep dividing until there is no remainder.

Example

Try it.

Divide: \(0.12\div 3.\)

Solution

	\(0.12\div 3\)
Write as long division, placing the decimal point in the quotient above the decimal point in the dividend.
Divide as usual. Since 3 does not go into 0 or 1 we use zeros as placeholders.
	\(0.12\div 3=0.04\)

Example

Try it.

Divide: \(\text{\$3.99}\div 24.\)

Solution

	\(\$3.99\div 24\)
Place the decimal point in the quotient above the decimal point in the dividend.
Divide as usual. When do we stop? Since this division involves money, we round it to the nearest cent (hundredth). To do this, we must carry the division to the thousandths place.
Round to the nearest cent.	\(\$0.166\approx \$0.17\)
	\(\$3.99\div 24\approx \$0.17\)

This means the price per bottle is \(17\) cents.

Condensed — the full section is in OpenStax Prealgebra 2e.

Use Decimals in Money Applications

We often apply decimals in real life, and most of the applications involving money. The Strategy for Applications we used in The Language of Algebra gives us a plan to follow to help find the answer. Take a moment to review that strategy now.

Example

Try it.

Paul received \(\text{\$50}\) for his birthday. He spent \(\text{\$31.64}\) on a video game. How much of Paul’s birthday money was left?

Solution

What are you asked to find?	How much did Paul have left?
Write a phrase.	$50 less $31.64
Translate.	\(50-31.64\)
Simplify.	18.36
Write a sentence.	Paul has $18.36 left.

Example

Try it.

Jessie put \(8\) gallons of gas in her car. One gallon of gas costs \(\text{\$3.529}.\) How much does Jessie owe for the gas? (Round the answer to the nearest cent.)

Solution

What are you asked to find?	How much did Jessie owe for all the gas?
Write a phrase.	8 times the cost of one gallon of gas
Translate.	\(8(\$3.529)\)
Simplify.	$28.232
Round to the nearest cent.	$28.23
Write a sentence.	Jessie owes $28.23 for her gas purchase.

Example

Try it.

Four friends went out for dinner. They shared a large pizza and a pitcher of soda. The total cost of their dinner was \(\text{\$31.76}.\) If they divide the cost equally, how much should each friend pay?

Solution

What are you asked to find?	How much should each friend pay?
Write a phrase.	$31.76 divided equally among the four friends.
Translate to an expression.	\(\$31.76\div 4\)
Simplify.	$7.94
Write a sentence.	Each friend should pay $7.94 for his share of the dinner.

Be careful to follow the order of operations in the next example. Remember to multiply before you add.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

• Add or subtract decimals.
  1. Write the numbers vertically so the decimal points line up.
  2. Use zeros as place holders, as needed.
  3. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.
• Multiply decimal numbers.
  1. Determine the sign of the product.
  2. Write the numbers in vertical format, lining up the numbers on the right.
  3. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
  4. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.
  5. Write the product with the appropriate sign.
• Multiply a decimal by a power of 10.
  1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
  2. Write zeros at the end of the number as placeholders if needed.
• Divide a decimal by a whole number.
  1. Write as long division, placing the decimal point in the quotient above the decimal point in the dividend.
  2. Divide as usual.
• Divide decimal numbers.
  1. Determine the sign of the quotient.
  2. Make the divisor a whole number by moving the decimal point all the way to the right. Move the decimal point in the dividend the same number of places to the right, writing zeros as needed.
  3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
  4. Write the quotient with the appropriate sign.
• Strategy for Applications
  1. Identify what you are asked to find.
  2. Write a phrase that gives the information to find it.
  3. Translate the phrase to an expression.
  4. Simplify the expression.
  5. Answer the question with a complete sentence.

Decimal Operations

Add and Subtract Decimals

In the following exercises, add or subtract.

Try it.

\(16.92+7.56\)

Solution

24.48

Try it.

\(18.37+9.36\)

Try it.

\(256.37-85.49\)

Solution

170.88

Try it.

\(248.25-91.29\)

Try it.

\(21.76-30.99\)

Solution

−9.23

Try it.

\(15.35-20.88\)

Try it.

\(37.5+12.23\)

Solution

49.73

Try it.

\(38.6+13.67\)

Try it.

\(-16.53-24.38\)

Solution

−40.91

Try it.

\(-19.47-32.58\)

Try it.

\(-38.69+31.47\)

Solution

−7.22

Try it.

\(-29.83+19.76\)

Try it.

\(-4.2+(-9.3)\)

Solution

−13.5

Try it.

\(-8.6+(-8.6)\)

Try it.

\(100-64.2\)

Solution

35.8

Try it.

\(100-65.83\)

Try it.

\(72.5-100\)

Solution

−27.5

Try it.

\(86.2-100\)

Try it.

\(15+0.73\)

Solution

15.73

Try it.

\(27+0.87\)

Try it.

\(2.51+40\)

Solution

42.51

Try it.

\(9.38+60\)

Try it.

\(91.75-(-10.462)\)

Solution

102.212

Try it.

\(94.69-(-12.678)\)

Try it.

\(55.01-3.7\)

Solution

51.31

Try it.

\(59.08-4.6\)

Try it.

\(2.51-7.4\)

Solution

−4.89

Try it.

\(3.84-6.1\)

Multiply Decimals

In the following exercises, multiply.

Try it.

\((0.3)(0.4)\)

Solution

0.12

Try it.

\((0.6)(0.7)\)

Try it.

\((0.24)(0.6)\)

Solution

0.144

Try it.

\((0.81)(0.3)\)

Try it.

\((5.9)(7.12)\)

Solution

42.008

Try it.

\((2.3)(9.41)\)

Try it.

\((8.52)(3.14)\)

Solution

26.7528

Try it.

\((5.32)(4.86)\)

Try it.

\(\text{(-4.3)(2.71)}\)

Solution

−11.653

Try it.

\((-8.5)(1.69)\)

Try it.

\(\text{(-5.18)(-65.23)}\)

Solution

337.8914

Try it.

\((-9.16)(-68.34)\)

Try it.

\((0.09)(24.78)\)

Solution

2.2302

Try it.

\((0.04)(36.89)\)

Try it.

\((0.06)(21.75)\)

Solution

1.305

Try it.

\((0.08)(52.45)\)

Try it.

\((9.24)(10)\)

Solution

92.4

Try it.

\((6.531)(10)\)

Try it.

\((55.2)(1,000)\)

Solution

55,200

Try it.

\((99.4)(1,000)\)

Divide Decimals

In the following exercises, divide.

Try it.

\(0.15\div 5\)

Solution

0.03

Try it.

\(0.27\div 3\)

Try it.

\(4.75\div 25\)

Solution

0.19

Try it.

\(12.04\div 43\)

Try it.

\(\text{\$8.49}\div 12\)

Solution

$0.71

Try it.

\(\text{\$16.99}\div 9\)

Try it.

\(\text{\$117.25}\div 48\)

Solution

$2.44

Try it.

\(\text{\$109.24}\div 36\)

Try it.

\(0.6\div 0.2\)

Solution

3

Try it.

\(0.8\div 0.4\)

Try it.

\(1.44\div (-0.3)\)

Solution

−4.8

Try it.

\(1.25\div (-0.5)\)

Try it.

\(-1.75\div (-0.05)\)

Solution

35

Try it.

\(-1.15\div (-0.05)\)

Try it.

\(5.2\div 2.5\)

Solution

2.08

Try it.

\(6.5\div 3.25\)

Try it.

\(12\div 0.08\)

Solution

150

Try it.

\(5\div 0.04\)

Try it.

\(11\div 0.55\)

Solution

20

Try it.

\(14\div 0.35\)

Mixed Practice

In the following exercises, simplify.

Try it.

\(6(12.4-9.2)\)

Solution

19.2

Try it.

\(3(15.7-8.6)\)

Try it.

\(24(0.5)+{(0.3)}^{2}\)

Solution

12.09

Try it.

\(35(0.2)+{(0.9)}^{2}\)

Try it.

\(1.15(26.83+1.61)\)

Solution

32.706

Try it.

\(1.18(46.22+3.71)\)

Try it.

\(\text{\$45}+0.08(\text{\$45})\)

Solution

$48.60

Try it.

\(\text{\$63}+0.18(\text{\$63})\)

Try it.

\(18\div (0.75+0.15)\)

Solution

20

Try it.

\(27\div (0.55+0.35)\)

Try it.

\((1.43+0.27)\div (0.9-0.05)\)

Solution

2

Try it.

\((1.5-0.06)\div (0.12+0.24)\)

Try it.

\([\text{\$75.42}+0.18(\text{\$75.42})]\div 5\)

Solution

$17.80

Try it.

\([\text{\$56.31}+0.22(\text{\$56.31})]\div 4\)

Use Decimals in Money Applications

In the following exercises, use the strategy for applications to solve.

Try it.

Spending money Brenda got \(\text{\$40}\) from the ATM. She spent \(\text{\$15.11}\) on a pair of earrings. How much money did she have left?

Solution

$24.89

Try it.

Spending money Marissa found \(\text{\$20}\) in her pocket. She spent \(\text{\$4.82}\) on a smoothie. How much of the \(\text{\$20}\) did she have left?

Try it.

Shopping Adam bought a t-shirt for \(\text{\$18.49}\) and a book for \(\text{\$8.92}\) The sales tax was \(\text{\$1.65}.\) How much did Adam spend?

Solution

$29.06

Try it.

Restaurant Roberto’s restaurant bill was \(\text{\$20.45}\) for the entrée and \(\text{\$3.15}\) for the drink. He left a \(\text{\$4.40}\) tip. How much did Roberto spend?

Try it.

Coupon Emily bought a box of cereal that cost \(\text{\$4.29}.\) She had a coupon for \(\text{\$0.55}\) off, and the store doubled the coupon. How much did she pay for the box of cereal?

Solution

$3.19

Try it.

Coupon Diana bought a can of coffee that cost \(\text{\$7.99}.\) She had a coupon for \(\text{\$0.75}\) off, and the store doubled the coupon. How much did she pay for the can of coffee?

Try it.

Diet Leo took part in a diet program. He weighed \(190\) pounds at the start of the program. During the first week, he lost \(4.3\) pounds. During the second week, he had lost \(2.8\) pounds. The third week, he gained \(0.7\) pounds. The fourth week, he lost \(1.9\) pounds. What did Leo weigh at the end of the fourth week?

Solution

181.7 pounds

Try it.

Snowpack On April \(1,\) the snowpack at the ski resort was \(4\) meters deep, but the next few days were very warm. By April \(5,\) the snow depth was \(1.6\) meters less. On April \(8,\) it snowed and added \(2.1\) meters of snow. What was the total depth of the snow?

Try it.

Coffee Noriko bought \(4\) coffees for herself and her co-workers. Each coffee was \(\text{\$3.75}.\) How much did she pay for all the coffees?

Solution

$15.00

Try it.

Subway Fare Arianna spends \(\text{\$4.50}\) per day on subway fare. Last week she rode the subway \(6\) days. How much did she spend for the subway fares?

Try it.

Income Mayra earns \(\text{\$9.25}\) per hour. Last week she worked \(32\) hours. How much did she earn?

Solution

$296.00

Try it.

Income Peter earns \(\text{\$8.75}\) per hour. Last week he worked \(19\) hours. How much did he earn?

Try it.

Hourly Wage Alan got his first paycheck from his new job. He worked \(30\) hours and earned \(\text{\$382.50}.\) How much does he earn per hour?

Solution

$12.75

Try it.

Hourly Wage Maria got her first paycheck from her new job. She worked \(25\) hours and earned \(\text{\$362.50}.\) How much does she earn per hour?

Try it.

Restaurant Jeannette and her friends love to order mud pie at their favorite restaurant. They always share just one piece of pie among themselves. With tax and tip, the total cost is \(\text{\$6.00}.\) How much does each girl pay if the total number sharing the mud pie is

ⓐ \(\ 2?\)

ⓑ \(\ 3?\)

ⓒ \(\ 4?\)

ⓓ \(\ 5?\)

ⓔ \(\ 6?\)

Solution

1. ⓐ $3
2. ⓑ $2
3. ⓒ $1.50
4. ⓓ $1.20
5. ⓔ $1

Try it.

Pizza Alex and his friends go out for pizza and video games once a week. They share the cost of a \(\text{\$15.60}\) pizza equally. How much does each person pay if the total number sharing the pizza is

ⓐ \(\ 2?\)

ⓑ \(\ 3?\)

ⓒ \(\ 4?\)

ⓓ \(\ 5?\)

ⓔ \(\ 6?\)

Try it.

Fast Food At their favorite fast food restaurant, the Carlson family orders \(4\) burgers that cost \(\text{\$3.29}\) each and \(2\) orders of fries at \(\text{\$2.74}\) each. What is the total cost of the order?

Solution

$18.64

Try it.

Home Goods Chelsea needs towels to take with her to college. She buys \(2\) bath towels that cost \(\text{\$9.99}\) each and \(6\) washcloths that cost \(\text{\$2.99}\) each. What is the total cost for the bath towels and washcloths?

Try it.

Zoo The Lewis and Chousmith families are planning to go to the zoo together. Adult tickets cost \(\text{\$29.95}\) and children’s tickets cost \(\text{\$19.95}.\) What will the total cost be for \(4\) adults and \(7\) children?

Solution

$259.45

Try it.

Ice Skating Jasmine wants to have her birthday party at the local ice skating rink. It will cost \(\text{\$8.25}\) per child and \(\text{\$12.95}\) per adult. What will the total cost be for \(12\) children and \(3\) adults?

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Simplify \(\frac{70}{100}.\)
   If you missed this problem, review .

   جواب کھوليں

   \(\frac{7}{10}\)

2. Multiply \(\frac{3}{10}\cdot \frac{9}{10}.\)
   If you missed this problem, review .

   جواب کھوليں

   \(\frac{27}{100}\)

3. Divide \(-36\div (-9).\)
   If you missed this problem, review .

   جواب کھوليں

   \(4\)

4. Add: \(3.7+12.4.\)

   جواب کھوليں

   	\(3.7+12.4\)
   Write the numbers vertically so the decimal points line up.	\(\begin{array}{l}3.7 \\ \underset{\text{_____}}{+12.4}\end{array}\)
   Place holders are not needed since both numbers have the same number of decimal places.
   Add the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.	\(\begin{array}{l}\overset{1}{3}.7 \\ \underset{\text{_____}}{+12.4} \\ 16.1\end{array}\)

5. Add: \(5.7+11.9.\)

   جواب کھوليں

   17.6

6. Add: \(18.32+14.79.\)

   جواب کھوليں

   33.11

7. Add: \(23.5+41.38.\)

   جواب کھوليں

   	\(23.5+41.38\)
   Write the numbers vertically so the decimal points line up.
   Place 0 as a place holder after the 5 in 23.5, so that both numbers have two decimal places.
   Add the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.

8. Add: \(4.8+11.69.\)

   جواب کھوليں

   16.49

9. Add: \(5.123+18.47.\)

   جواب کھوليں

   23.593

10. Subtract: \(20-14.65.\)

    جواب کھوليں

    	\(20-14.65\)
    Write the numbers vertically so the decimal points line up. Remember 20 is a whole number, so place the decimal point after the 0.
    Place two zeros after the decimal point in 20, as place holders so that both numbers have two decimal places.
    Subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.

11. Subtract:

    \(10-9.58.\)

    جواب کھوليں

    0.42

12. Subtract:

    \(50-37.42.\)

    جواب کھوليں

    12.58

13. Subtract: \(2.51-7.4.\)

    جواب کھوليں

    If we subtract \(7.4\) from \(2.51,\) the answer will be negative since \(7.4>2.51.\) To subtract easily, we can subtract \(2.51\) from \(7.4.\) Then we will place the negative sign in the result.
    

    	\(2.51-7.4\)
    Write the numbers vertically so the decimal points line up.
    Place zero after the 4 in 7.4 as a place holder, so that both numbers have two decimal places.
    Subtract and place the decimal in the answer.
    Remember that we are really subtracting \(2.51-7.4\) so the answer is negative.	\(2.51-7.4=-4.89\)

14. Subtract: \(4.77-6.3.\)

    جواب کھوليں

    −1.53

15. Subtract: \(8.12-11.7.\)

    جواب کھوليں

    −3.58

16. Multiply: \((3.9)(4.075).\)

    جواب کھوليں

    	\((3.9)(4.075)\)
    Determine the sign of the product. The signs are the same.	The product will be positive.
    Write the numbers in vertical format, lining up the numbers on the right.
    Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
    Place the decimal point. Add the number of decimal places in the factors \((1+3).\) Place the decimal point 4 places from the right.
    The product is positive.	\((3.9)(4.075)=15.8925\)

17. Multiply: \(4.5(6.107).\)

    جواب کھوليں

    27.4815

18. Multiply: \(10.79(8.12).\)

    جواب کھوليں

    87.6148

19. Multiply: \((-8.2)\text{(}5.19\text{).}\)

    جواب کھوليں

    	\((-8.2)(5.19)\)
    The signs are different.	The product will be negative.
    Write in vertical format, lining up the numbers on the right.	\(\begin{array}{l}5.19 \\ \underset{\text{_____}}{\times 8.2}\end{array}\)
    Multiply.	\(\begin{array}{l}5.19 \\ \underset{\text{_____}}{\times 8.2} \\ 1038 \\ \underset{\text{_____}}{4152} \\ 42558\end{array}\)
    	\(\begin{array}{l}5.19 \\ \underset{\text{_____}}{\times 8.2} \\ 1038 \\ \underset{\text{_____}}{4152} \\ 42.558\end{array}\)
    The product is negative.	\((-8.2)(5.19)=-42.558\)

20. Multiply: \((4.63)(-2.9)\text{.}\)

    جواب کھوليں

    −13.427

21. Multiply: \((-7.78)(4.9)\text{.}\)

    جواب کھوليں

    −38.122

22. Multiply: \((0.03)\text{(}0.045\text{).}\)

    جواب کھوليں

    	\((0.03)(0.045)\)
    The product is positive.
    Write in vertical format, lining up the numbers on the right.
    Multiply.
    Add zeros as needed to get the 5 places.
    The product is positive.	\((0.03)(0.045)=0.00135\)

23. Multiply: \((0.04)(0.087)\text{.}\)

    جواب کھوليں

    0.00348

24. Multiply: \((0.09)(0.067)\text{.}\)

    جواب کھوليں

    0.00603

25. Multiply \(5.63\) by factors of ⓐ \(\ 10\) ⓑ \(\ 100\ \\)ⓒ \(\ 1000.\)

    جواب کھوليں

    By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.

    ⓐ
    	\(56.3(10)\)
    There is 1 zero in 10, so move the decimal point 1 place to the right.
    	\(56.3\)

    ⓑ
    	\(5.63(100)\)
    There are 2 zeros in 100, so move the decimal point 2 places to the right.
    	\(563\)

    ⓒ
    	\(5.63(1000)\)
    There are 3 zeros in 1000, so move the decimal point 3 places to the right.
    A zero must be added at the end.	\(5,630\)

26. Multiply \(2.58\) by factors of ⓐ \(\ 10\) ⓑ \(\ 100\) ⓒ \(\ 1000.\)

    جواب کھوليں

    1. ⓐ 25.8
    2. ⓑ 258
    3. ⓒ 2,580

27. Multiply \(14.2\) by factors of ⓐ \(\ 10\) ⓑ \(\ 100\) ⓒ \(\ 1000.\)

    جواب کھوليں

    1. ⓐ 142
    2. ⓑ 1,420
    3. ⓒ 14,200

28. Divide: \(0.12\div 3.\)

    جواب کھوليں

    	\(0.12\div 3\)
    Write as long division, placing the decimal point in the quotient above the decimal point in the dividend.
    Divide as usual. Since 3 does not go into 0 or 1 we use zeros as placeholders.
    	\(0.12\div 3=0.04\)

29. Divide: \(0.28\div 4.\)

    جواب کھوليں

    0.07

30. Divide: \(0.56\div 7.\)

    جواب کھوليں

    0.08

31. Divide: \(\text{\$3.99}\div 24.\)

    جواب کھوليں

    	\(\$3.99\div 24\)
    Place the decimal point in the quotient above the decimal point in the dividend.
    Divide as usual. When do we stop? Since this division involves money, we round it to the nearest cent (hundredth). To do this, we must carry the division to the thousandths place.
    Round to the nearest cent.	\(\$0.166\approx \$0.17\)
    	\(\$3.99\div 24\approx \$0.17\)

    This means the price per bottle is \(17\) cents.

32. Divide: \(\text{\$6.99}\div 36.\)

    جواب کھوليں

    $0.19

33. Divide: \(\text{\$4.99}\div 12.\)

    جواب کھوليں

    $0.42

34. Divide: \(-2.89\div \text{(}3.4\text{)}.\)

    جواب کھوليں

    Determine the sign of the quotient.	The quotient will be negative.
    Make the divisor the whole number by 'moving' the decimal point all the way to the right. 'Move' the decimal point in the dividend the same number of places to the right.
    Divide. Place the decimal point in the quotient above the decimal point in the dividend. Add zeros as needed until the remainder is zero.
    Write the quotient with the appropriate sign.	\(-2.89\div (3.4)=-0.85\)

35. Divide: \(-1.989\div 5.1.\)

    جواب کھوليں

    −0.39

36. Divide: \(-2.04\div 5.1.\)

    جواب کھوليں

    −0.4

37. Divide: \(-25.65\div \text{(}-0.06\text{).}\)

    جواب کھوليں

    	\(-25.65\div (-0.06)\)
    The signs are the same.	The quotient is positive.
    Make the divisor a whole number by 'moving' the decimal point all the way to the right. 'Move' the decimal point in the dividend the same number of places.
    Divide. Place the decimal point in the quotient above the decimal point in the dividend.
    Write the quotient with the appropriate sign.	\(-25.65\div (-0.06)=427.5\)

38. Divide: \(-23.492\div (-0.04)\text{.}\)

    جواب کھوليں

    587.3

39. Divide: \(-4.11\div (-0.12)\text{.}\)

    جواب کھوليں

    34.25

40. Divide: \(4\div 0.05.\)

    جواب کھوليں

    	\(4\div 0.05\)
    The signs are the same.	The quotient is positive.
    Make the divisor a whole number by 'moving' the decimal point all the way to the right. Move the decimal point in the dividend the same number of places, adding zeros as needed.
    Divide. Place the decimal point in the quotient above the decimal point in the dividend.
    Write the quotient with the appropriate sign.	\(4\div 0.05=80\)

    We can relate this example to money. How many nickels are there in four dollars? Because \(4\div 0.05=80,\) there are \(80\) nickels in \(\text{\$4}.\)

Symbols used here

\approx

approximately equal
Equal to the precision shown, not exactly.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

\%

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Decimal Operations

1. Add and subtract decimals
2. Multiply decimals
3. Divide decimals
4. Use decimals in money applications
5. Write the numbers vertically so the decimal points line up.
6. Use zeros as place holders, as needed.
7. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.
8. if their signs are the same, the product is positive.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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حل

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

میں زیادہ Arithmetic

Order of operationsFractionsPercentagesPowers and roots