Learning Objectives

- Use three different ways to represent division.
- Divide whole numbers.
- Perform long division.
- Divide whole numbers by a power of 10.
- Recognize that division by 0 is not defined.
- Solve application problems using division.

Some people think about division as “fair sharing” because when you divide a number you are trying to create equal parts. Division is also the **inverse operation** of multiplication because it “undoes” multiplication. In multiplication, you combine equal sets to create a total. In division, you separate a whole group into sets that have the same amount. For example, you could use division to determine how to share 40 empanadas among 12 guests at a party.

Division is splitting into equal parts or groups. For example, one might use division to determine how to share a plate of cookies evenly among a group. If there are 15 cookies to be shared among five people, you could divide 15 by 5 to find the “fair share” that each person would get. Consider the picture below.

15 cookies split evenly across 5 plates results in 3 cookies on each plate. You could represent this situation with the equation:

( 15 div 5=3)

You could also use a number line to model this division. Just as you can think of multiplication as repeated addition, you can think of division as repeated subtraction. Consider how many jumps you take by 5s as you move from 15 back to 0 on the number line.

Notice that there are 3 jumps that you make when you skip count by 5 from 15 back to 0 on the number line. This is like subtracting 5 from 15 three times. This repeated subtraction can be represented by the equation: ( 15 div 5=3).

Finally, consider how an area model can show this division. Ask yourself, if you were to make a rectangle that contained 15 squares with 5 squares in a row, how many rows would there be in the rectangle? Start by making one row of 5 squares:

Then add two more rows of 5 squares so you have 15 squares.

The number of rows is 3. So, 15 divided by 5 is equal to 3.

Example

**Find ( 24 div 3) using a set model and a number line model.**

**Solution**

Set Model:

Number line model:

( 24 div 3=8)

As with multiplication, division can be written using a few different symbols. We showed this division written as ( 15 div 5=3), but it can also be written two other ways:

( egin{array}{r}

3\

5 longdiv { 1 5 }

end{array})

( frac{15}{5}=3)

Each part of a division problem has a name. The number that is being divided up, that is the total, is called the **dividend**. In the work in this topic, this number will be the larger number, but that is not always true in mathematics. The number that is dividing the dividend is called the **divisor**. The answer to a division problem is called the **quotient**.

The blue box below summarizes the terminology and common ways to represent division.

Three Ways to Represent Division

( 12 div 3=4) (**with a division symbol**; this equation is read "12 divided by 3 equals 4."

( egin{array}{r}

4\

3 longdiv { 1 2 }

end{array}) (**with a division or long division symbol**; this expression is read "12 divided by 3 equals 4." Notice here, though, that you have to start with what is underneath the symbol. This may take some getting used to since you are reading from right to left and bottom to top!)

( frac{12}{3}=4) (**with a fraction bar**; this expression can also be read "12 divided by 3 equals 4." In this format, you read from top to bottom.)

In the examples above, 12 is the** dividend**, 3 is the **divisor **and 4 is the** quotient.**

( ext { Dividend } div ext { Divisor }= ext { Quotient })

( egin{array}{r}

ext{Quotient}

ext{Divisor}longdiv{ ext{Dividend}}

end{array})

( frac{ ext { Dividend }}{ ext { Divisor }}= ext { Quotient })

Exercise

Which of the following expressions represent dividing $56 equally among 7 people?

#1: ( frac{7}{56})

#2: ( 56 div 7)

#3: ( 56longdiv {7})

- #2 represents the situation.
- All three expressions represent the situation.
- #1 represents the situation.
- #3 represents the situation.

**Answer**#2 represents the situation.

Correct. #2 is the only expression that represents 56 divided by 7.

All three expressions represent the situation.

Incorrect. #1 and #3 represent 7 divided by 56, not 56 divided by 7. The correct answer is #2 is the only expression that represents the situation.

#1 represents the situation.

Incorrect. This expression represents 7 divided by 56, not 56 divided by 7. The correct answer is #2 is the only expression that represents the situation.

#3 represents the situation.

Incorrect. This expression represents 7 divided by 56, not 56 divided by 7. The correct answer is #2 is the only expression that represents the situation.

Once you understand how division is written, you are on your way to solving simple division problems. You will need your multiplication facts to perform division. If you do not have them memorized, you can guess and check or use a calculator.

Consider the following problems:

( 10 div 5=?)

( 48 div 2=?)

( 30 div 5=?)

In the first problem, ( 10 div 5), you could ask yourself, “how many fives are there in ten?” You can probably answer this easily. Another way to think of this is to consider breaking up 10 into 5 groups and picturing how many would be in each group.

( 10 div 5=2)

To solve ( 48 div 2), you might realize that dividing by 2 is like splitting into two groups or splitting the total in half. What number could you double to get 48?

( 48 div 2=24)

To figure out ( 30 div 5), you could ask yourself, how many times do you have to skip count by 5 to get from 0 to 30? 5, 10, 15, 20, 25, 30. You have to skip count 6 times to get to 30.

( 30 div 5=6)

Exercise

Compute ( 35 div 5).

**Answer**7

Exercise

Compute ( 32 div 4).

**Answer**8

Sometimes when you are dividing, you cannot easily share the number equally. Think about the division problem ( 9 div 2). You could think of this problem as 9 pieces of chocolate being split between 2 people. You could make two groups of 4 chocolates, and you would have one chocolate left over.

In mathematics, this left over part is called the **remainder**. It is the part that *remains *after performing the division. In the example above, the remainder is 1. We can write this as:

( 9 div 2=4quad mathrm{R} 1)

We read this equation: “Nine divided by two equals four with a remainder of 1."

You might be thinking you could split that extra piece of chocolate in parts to share it. This is great thinking! If you split the chocolate in half, you could give each person another half of a piece of chocolate. They would each get ( 4 frac{1}{2}) pieces of chocolate. We are not going to worry about expressing remainders as fractions or decimals right now. We are going to use the remainder notation with the letter R. Here’s an example:

Example

( 45 div 6)

**Solution**

( 6 cdot 7=42) | How many sixes are there in 45? Try 7. |

( 45-42=3) | 3 is not enough for another 6. So, 3 is the remainder. |

( 45 div 6=7quad mathrm{R} 3)

Since multiplication is the inverse of division, you can check your answer to a division problem with multiplication. To check the answer 7 R3, first multiply 6 by 7 and then add 3.

( 6 cdot 7=42)

( 42+3=45), so the quotient 7 R3 is correct.

Exercise

Compute ( 67 div 7).

- 9
- 9 R4
- 60
- 10

**Answer**9

Incorrect. ( 9 cdot 7=63). There is a remainder of 4. The correct answer is 9 R4.

9 R4

Correct. ( 9 cdot 7=63) and there are 4 left over.

60

Incorrect. This is a division, not subtraction, problem. The correct answer is 9 R4.

10

Incorrect. ( 70 div 7=10), so the answer to ( 67 div 7) cannot be; ( 9 cdot 7=63) and there are 4 left over. The correct answer is 9 R4.

Long division is a method that is helpful when you are performing division that you cannot do easily in your head, such as division involving larger numbers. Below is an example of a way to write out the division steps.

Example

( 68 div 4)

**Solution**

( 4 longdiv { 6 8 }) | Rewrite the division. |

( egin{array}{r} 1\ 4longdiv{68} -4 \ hline 28 end{array}) | Divide the tens. What is 6 divided by 4? ( 4 cdot 1=4), so write a 1 above the 6. Subtract 4 from 6 and bring down the next digit of the dividend, 8. What is 28 divided by 4? |

( egin{array}{r} 17\ 4longdiv{68} -4 \ hline 28\ -28\ hline 0 end{array}) | ( 7 cdot 4=28), so write a 7 above the 8. There is no remainder. |

( egin{array}{r} ^217 \ | Check your answer using multiplication. |

( 68 div 4=17)

Example

( 6,707 div 233)

**Solution**

( egin{array}{r} 2\ 233longdiv{6707} -466 \ hline end{array}) | Examine the first 3 digits of the dividend and determine how many 233s are in it. Use guess and check. Try: ( 2 cdot 233=466) Try: ( 3 cdot 233=699) (too large) |

( egin{array}{r} 2\ 233longdiv{6707} -466 \ hline 2047 end{array}) | Subtract 466 from 670 and bring down the next digit of the dividend, 7. |

( egin{array}{r} 28\ 233longdiv{6707} -466 \ hline 2047 \ -1864 \ hline 183 end{array}) | How many 233s are in 2,047? It looks like close to 10 because ( 233 cdot 10=2,330). Try ( 9.233 cdot 9) equals 2,097. ( egin{array}{r} 2,097 (Too large) Must be 8! ( egin{array}{r} |

( 233 cdot 28=6,524) ( 6,524+183=6,707) | Check your answer using multiplication. First, multiply ( 233 cdot 28). Then, add the remainder. |

( 6,707 div 233=28 quadmathrm{R} 183)

Exercise

Compute ( 417 div 34).

- 451
- 12
- 12 R9
- 13

**Answer**451

Incorrect. This is a division problem, not an addition problem. The correct answer is 12 R9.

12

Incorrect. ( 12 cdot 34=408). The correct answer is 12 R9.

12 R9

Correct. ( 12 cdot 34=408) and ( 408+9=417)

13

Incorrect. ( 13 cdot 34=442). The correct answer is 12 R9.

Just as multiplication by powers of 10 results in a pattern, there is a pattern with division by powers of 10. Consider three quotients: ( 20 div 10 ; 200 div 10 ; 2,000 div 10).

Think about ( 20 div 10). There are 2 tens in twenty, so ( 20 div 10=2). The computations for ( 200 div 10) and ( 2,000 div 10) are shown below.

Example

( 200 div 10)

**Solution**

( 1 0 longdiv { 2 0 0 }) | Rewrite the problem. |

( 1 0 longdiv { {color{red}2} {color{blue}0} 0 }) | Divide the first digit of the dividend, 2, by the divisor. Since ( 2 div 10) does not give a whole number, go to the next digit, 0. |

( egin{array}{r} 2\ 1 0 longdiv { {color{red}2 0} 0 } end{array}) | ( 20 div 10=2) |

( egin{array}{r} 2\ 10longdiv{200} -20 \ hline 0 end{array}) | ( 2 cdot 10=20) ( 20-20=0) |

( egin{array}{r} 2\ 10longdiv{20color{red}0} -20 \ hline 0color{red}0 end{array}) | Bring down the next digit of the dividend, which is 0. |

( egin{array}{r} 2color{blue}0 10longdiv{200} -20 \ hline 00 \ 0 \ hline 0 end{array}) | Since 10 still does not go into 00 and we have nothing left to bring down, multiply the 0 by 10. ( 0 cdot 10=0) ( 0-0=0) We have no remainder. |

( 200 div 10=20)

Example

( 2000 div 10)

**Solution**

( 1 0 longdiv { 2 0 0 0 }) | Rewrite the problem. |

( 1 0 longdiv { {color{red}2} 0 0 0 }) | Divide the first digit of the dividend, 2, by the divisor. Since ( 2 div 10) does not give a whole number, go to the next digit, 0. |

( egin{array}{r} 2\ 1 0 longdiv { {color{red}2 0} 0 0 } end{array}) | ( 20 div 10=2) |

( egin{array}{r} 2\ 10longdiv{2000} -20 \ hline 0 end{array}) | ( 2 cdot 10=20) ( 20-20=0) |

( egin{array}{r} 2\ 10longdiv{20{color{red}0}0} -20 \ hline 0color{red}0 end{array}) | Bring down the next digit, 0, of the dividend. |

( egin{array}{r} 2color{blue}0 10longdiv{200color{red}0} -20 \ hline 00color{red}0 end{array}) | Since 10 does not go into 00, add a 0 to the quotient and bring down the next digit, 0. |

( egin{array}{r} 20color{blue}0 10longdiv{2000} -20 \ hline 000\ 0\ hline 0 end{array}) | Since 10 still does not go into 000 and we have nothing left to bring down, add a 0 to the quotient, multiply the 0 by 10. ( 0 cdot 10=0) ( 0-0=0) We have no remainder. |

( 2,000 div 10=200)

Examine the results of these three problems to try to determine a pattern in division by 10.

( egin{aligned}

20 div 10 &=2

200 div 10 &=20

2,000 div 10 &=200

end{aligned})

Notice that the number of zeros in the quotient decreases when a dividend is divided by 10: 20 becomes 2; 200 becomes 20 and 2,000 become 200. In each of the examples above, you can see that there is one fewer 0 in the quotient than there was in the dividend.

Continue another example of division by a power of 10.

Example

( 2,000 div 100)

**Solution**

( 1 0 0 longdiv { 2 0 0 0 }) | Rewrite the problem. |

( 1 0 0 longdiv { {color{red}2} 0 0 0 }) | Divide the first digit of the dividend, 2, by the divisor. Since ( 2 div 100) does not give a whole number, go to the next digit, 0. |

( 1 0 0 longdiv { {color{red}2 0} 0 0 }) | Divide the first two digits of the dividend, 20, by the divisor. Since ( 20 div 100) does not give a whole number, go to the next digit, 0. |

( 1 0 0 longdiv { {color{red}2 0 0} 0 }) | ( 200 div 100=2) |

( egin{array}{r} 2\ 100longdiv{2000} -200 \ hline 0 end{array}) | ( 2 cdot 100=200) ( 200-200=0) |

( egin{array}{r} 2\ 100longdiv{200color{red}0} -200 \ hline 0color{red}0 end{array}) | Bring down the next digit, 0, of the dividend. |

( egin{array}{r} 2color{blue}0 100longdiv{2000} -200 \ hline 00\ 0\ hline 0 end{array}) | Since 100 still does not go into 00 and we have nothing left to bring down, add a 0 to the quotient, multiply the 0 by 10. ( 0 cdot 10=0) ( 0-0=0) We have no remainder. |

( 2,000 div 100=20)

Consider this set of examples of division by powers of 10. What pattern do you see?

( egin{array}{rl}

20 &div &10=2

200 &div &10=20

2,000 &div &10=200

2,000 &div &100=20

2,000 &div &1,000=2

end{array})

Notice that when you divide a number by a power of 10, the quotient has fewer zeros. This is because division by a power of 10 has an effect on the place value. For example, when you perform the division ( 18,000 div 100=180), the quotient, 180, has two fewer zeros than the dividend, 18,000. This is because the power of 10 divisor, 100, has two zeros.

Exercise

Compute ( 135,000 div 100).

- 13,500
- 134,900
- 13,500,000
- 1,350

**Answer**13,500

Incorrect. This answer is too large. ( 13,500 cdot 100=1,350,000). The correct answer is 1,350.

134,900

Incorrect. This is a division, not a subtraction, problem. The correct answer is 1,350.

13,500,000

Incorrect. This is a division, not a multiplication, problem. The correct answer is 1,350.

1,350

Correct. 1,350 cdot 100=135,000.

You know what it means to divide by 2 or divide by 10, but what does it mean to divide a quantity by 0? Is this even possible? Can you divide 0 by a number? Consider the two problems written below.

( frac{0}{8} ext { and } frac{8}{0})

We can read the first expression, “zero divided by eight” and the second expression, “eight divided by zero.” Since multiplication is the inverse of division, we could rewrite these as multiplication problems.

( 0 div 8=?)

( ? cdot 8=0)

The quotient must be 0 because ( 0 cdot 8=0).

( frac{0}{8}=0)

Now let’s consider ( frac{8}{0}).

( 8 div 0=?)

( ? cdot 0=8)

This is not possible. There is no number that you could multiply by zero and get eight. Any number multiplied by zero is always zero. There is no quotient for ( frac{8}{0}). There is no quotient for any number when it is divided by zero.

Division by zero is an operation for which you cannot find an answer, so it is not allowed. We say that division by 0 is undefined.

Division is used in solving many types of problems. Below are three examples from real life that use division in their solutions.

Example

**Luana made 40 empanadas for a party. If the empanadas are divided equally among 12 guests, how many will each guest have? Will there be any leftover empanadas?**

**Solution**

( 40 div 12) | Since each guest will have an equal share, we can use division. |

( egin{array}{r} 3\ 12longdiv{40} -36\ hline 4 end{array}) | Use trial and error. Try 3. ( 12 cdot 3=36) When 40 empanadas are divided equally among 12 people, there are 4 left over. |

Each guest will have 3 empanadas. There will be 4 empanadas left over.

Example

**A case of floor tiles has 12 boxes in it. The case costs $384. How much does one box cost?**

**Solution**

( 384 div 12) | Since the boxes each cost the same amount, you want to divide $384 into 12 equal parts. |

( egin{array}{r} 12longdiv{384} end{array}) | Perform the division. Try to divide the first digit in the dividend by the divisor. 12 will not divide into 3, so go to the next digit. |

( egin{array}{r} 3\ 12longdiv{384} -36 \ hline end{array}) | Perform ( 38 div 12). Pick a quotient and test it. Try 3. ( 3 cdot 12=36). |

( egin{array}{r} 3\ 12longdiv{384} -36 \ hline 2 end{array}) | Subtract 36 from 38. |

( egin{array}{r} 32\ 12longdiv{384} -36 \ hline 24\ -24\ hline 0 end{array}) | Bring the next digit of the dividend down and perform division. ( 12 cdot 2=24) ( 24-24=0) |

( egin{array}{r} 32 \ imesquad 12 hline 64 +quad320 hline 384 end{array}) | Does ( 32 cdot 12) equal 384? Check your answer by multiplying. Yes! The answer is correct! |

Each box of tiles costs $32

Example

**A banana grower is shipping 4,644 bananas. There are 86 crates, each containing the same number of bananas. How many bananas are in each crate?**

**Solution**

( 4,644 div 86) | Since each crate, or box, has the same number of bananas, you can take the total number of bananas and divide by the number of crates. Rewrite the division. |

( 8 6 longdiv { 4 6 4 4 }) | Use trial and error to determine what ( 464 div 86) equals. Try 5: ( egin{array}{r} |

( egin{array}{r} 5\ 86longdiv{4644} -430 \ hline 344 end{array}) | ( 464-430=34) Then, bring down the next digit of the dividend, 4. |

( egin{array}{r} 54\ 86longdiv{4644} -430 \ hline 344\ -344\ hline 0 end{array}) | Use trial and error to determine the quotient of 344 and 86. Try 4: ( egin{array}{r} |

( egin{array}{r} 86 \ imes 54 hline 344 +4300 \ hline 4644 end{array}) | Check your answer by multiplying. Yes! The answer is correct! |

Each crate contains 54 bananas.

Exercise

A theater has 1,440 seats. The theater has 30 rows of seats. How many seats are in each row?

- 1,410
- 48
- 43,200
- 480

**Answer**1,410

Incorrect. Use division, ( 1440 div 30), not subtraction for this problem. The correct answer is 48.

48

Correct. ( 1440 div 30=48).

43,200

Incorrect. The answer is too large. Use division, ( 1440 div 30), not multiplication, for this problem. The correct answer is 48.

480

Incorrect. There is a place-value error. The correct answer is 48.

Division is the inverse operation of multiplication, and can be used to determine how to evenly share a quantity among a group. Division can be written in three different ways: using a fraction bar, using a division symbol, and using long division. Division can be represented as splitting a total quantity into sets of equal quantities, as skip subtracting on the number line, and as a dimension with an area model. Remainders may result when performing division and they can be represented with the letter R, followed by the number remaining. Since division is the inverse operation of multiplication, you need to know your multiplication facts in order to do division. For larger numbers, you can use long division to find the quotient.

## Golden ratio

In mathematics, two quantities are in the **golden ratio** if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities *a* and *b* with *a* > *b* > 0,

The golden ratio is also called the **golden mean** or **golden section** (Latin: *sectio aurea*). [4] [5] Other names include **extreme and mean ratio**, [6] **medial section**, **divine proportion** (Latin: *proportio divina*), [7] **divine section** (Latin: *sectio divina*), **golden proportion**, **golden cut**, [8] and **golden number**. [9] [10] [11]

Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data. [12] The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts.

Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing this to be aesthetically pleasing. These often appear in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio.

## 1.3 Subtract Whole Numbers

Suppose there are seven bananas in a bowl. Elana uses three of them to make a smoothie. How many bananas are left in the bowl? To answer the question, we subtract three from seven. When we subtract, we take one number away from another to find the difference . The notation we use to subtract 3 3 from 7 7 is

### Subtraction Notation

To describe subtraction, we can use symbols and words.

### Example 1.26

Translate from math notation to words: ⓐ 8 − 1 8 − 1 ⓑ 26 − 14 26 − 14 .

#### Solution

- ⓐ We read this as
*eight minus one*. The result is*the difference of eight and one*. - ⓑ We read this as
*twenty-six minus fourteen*. The resuilt is*the difference of twenty-six and fourteen*.

Translate from math notation to words:

Translate from math notation to words:

### Model Subtraction of Whole Numbers

We start by modeling the first number, 7. | |

Now take away the second number, 3. We'll circle 3 blocks to show that we are taking them away. | |

Count the number of blocks remaining. | |

There are 4 ones blocks left. | We have shown that 7 − 3 = 4 7 − 3 = 4 . |

### Manipulative Mathematics

### Example 1.27

Model the subtraction: 8 − 2 . 8 − 2 .

#### Solution

### Example 1.28

Model the subtraction: 13 − 8 . 13 − 8 .

#### Solution

Model the first number, 13. We use 1 ten and 3 ones. | |

Take away the second number, 8. However, there are not 8 ones, so we will exchange the 1 ten for 10 ones. | |

Now we can take away 8 ones. | |

Count the blocks remaining. | |

There are five ones left. | We have shown that 13 − 8 = 5 13 − 8 = 5 . |

As we did with addition, we can describe the models as ones blocks and tens rods, or we can simply say ones and tens.

Model the subtraction: 12 − 7 . 12 − 7 .

Model the subtraction: 14 − 8 . 14 − 8 .

### Example 1.29

Model the subtraction: 43 − 26 . 43 − 26 .

#### Solution

Count the number of blocks remaining. There is 1 1 ten and 7 7 ones, which is 17 . 17 .

Model the subtraction: 42 − 27 . 42 − 27 .

Model the subtraction: 45 − 29 . 45 − 29 .

### Subtract Whole Numbers

Addition and subtraction are inverse operations. Addition undoes subtraction, and subtraction undoes addition.

### Example 1.30

Subtract and then check by adding:

#### Solution

Subtract and then check by adding:

Subtract and then check by adding:

To subtract numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition. Align the digits by place value, and then subtract each column starting with the ones and then working to the left.

### Example 1.31

Subtract and then check by adding: 89 − 61 . 89 − 61 .

#### Solution

Subtract and then check by adding: 86 − 54 . 86 − 54 .

Subtract and then check by adding: 99 − 74 . 99 − 74 .

### How To

#### Find the difference of whole numbers.

- Step 1. Write the numbers so each place value lines up vertically.
- Step 2. Subtract the digits in each place value. Work from right to left starting with the ones place. If the digit on top is less than the digit below, borrow as needed.
- Step 3. Continue subtracting each place value from right to left, borrowing if needed.
- Step 4. Check by adding.

### Example 1.32

#### Solution

Write the numbers so each place value lines up vertically. |

Subtract the ones. We cannot subtract 6 from 3, so we borrow 1 ten. This makes 3 tens and 13 ones. We write these numbers above each place and cross out the original digits. |

Now we can subtract the ones. 13 − 6 = 7. 13 − 6 = 7. We write the 7 in the ones place in the difference. |

Now we subtract the tens. 3 − 2 = 1. 3 − 2 = 1. We write the 1 in the tens place in the difference. |

Check by adding. Our answer is correct. |

Subtract and then check by adding: 93 − 58 . 93 − 58 .

Subtract and then check by adding: 81 − 39 . 81 − 39 .

### Example 1.33

Subtract and then check by adding: 207 − 64 . 207 − 64 .

#### Solution

Subtract and then check by adding: 439 − 52 . 439 − 52 .

Subtract and then check by adding: 318 − 75 . 318 − 75 .

### Example 1.34

Subtract and then check by adding: 910 − 586 . 910 − 586 .

#### Solution

Write the numbers so each place value lines up vertically. |

Subtract the ones. We cannot subtract 6 from 0, so we borrow 1 ten and add 10 ones to the 10 ones we had. This makes 10 ones. We write a 0 above the tens place and cross out the 1. We write the 10 above the ones place and cross out the 0. Now we can subtract the ones. 10 − 6 = 4. 10 − 6 = 4. |

Write the 4 in the ones place of the difference. |

Subtract the tens. We cannot subtract 8 from 0, so we borrow 1 hundred and add 10 tens to the 0 tens we had, which gives us 10 tens. Write 8 above the hundreds place and cross out the 9. Write 10 above the tens place. |

Now we can subtract the tens. 10 − 8 = 2 10 − 8 = 2 . |

Subtract the hundreds place. 8 − 5 = 3 8 − 5 = 3 Write the 3 in the hundreds place in the difference. |

Check by adding. Our answer is correct. |

Subtract and then check by adding: 832 − 376 . 832 − 376 .

Subtract and then check by adding: 847 − 578 . 847 − 578 .

### Example 1.35

Subtract and then check by adding: 2,162 − 479 . 2,162 − 479 .

#### Solution

Write the numbers so each place values line up vertically. | |

Subtract the ones. Since we cannot subtract 9 from 2, borrow 1 ten and add 10 ones to the 2 ones to make 12 ones. Write 5 above the tens place and cross out the 6. Write 12 above the ones place and cross out the 2. | |

Now we can subtract the ones. | 12 − 9 = 3 12 − 9 = 3 |

Write 3 in the ones place in the difference. | |

Subtract the tens. Since we cannot subtract 7 from 5, borrow 1 hundred and add 10 tens to the 5 tens to make 15 tens. Write 0 above the hundreds place and cross out the 1. Write 15 above the tens place. | |

Now we can subtract the tens. | 15 − 7 = 8 15 − 7 = 8 |

Write 8 in the tens place in the difference. | |

Now we can subtract the hundreds. | |

Write 6 in the hundreds place in the difference. | |

Subtract the thousands. There is no digit in the thousands place of the bottom number, so we imagine a 0. 1 − 0 = 1. 1 − 0 = 1. Write 1 in the thousands place of the difference. | |

Check by adding. 1 1 , 6 1 8 1 3 + 479 ______ 2, 162 ✓ 1 1 , 6 1 8 1 3 + 479 ______ 2, 162 ✓ |

Subtract and then check by adding: 4,585 − 697 . 4,585 − 697 .

Subtract and then check by adding: 5,637 − 899 . 5,637 − 899 .

### Translate Word Phrases to Math Notation

As with addition, word phrases can tell us to operate on two numbers using subtraction. To translate from a word phrase to math notation, we look for key words that indicate subtraction. Some of the words that indicate subtraction are listed in Table 1.3.

### Example 1.36

Translate and then simplify:

#### Solution

The word *difference* tells us to subtract the two numbers. The numbers stay in the same order as in the phrase.

The words *subtract* *from* tells us to take the second number away from the first. We must be careful to get the order correct.

### Subtract Whole Numbers in Applications

To solve applications with subtraction, we will use the same plan that we used with addition. First, we need to determine what we are asked to find. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question, using the appropriate units.

### Example 1.37

#### Solution

We are asked to find the difference between the morning temperature and the noon temperature.

Write a phrase. | the difference of 73 and 27 |

Translate to math notation. Difference tells us to subtract. | 73 − 27 73 − 27 |

Then we do the subtraction. | |

Write a sentence to answer the question. | The difference in temperatures was 46 degrees Fahrenheit. |

### Example 1.38

#### Solution

We are asked to find the difference between the regular price and the sale price.

### Media

#### ACCESS ADDITIONAL ONLINE RESOURCES

### Section 1.3 Exercises

#### Practice Makes Perfect

**Use Subtraction Notation**

In the following exercises, translate from math notation to words.

**Model Subtraction of Whole Numbers**

In the following exercises, model the subtraction.

**Subtract Whole Numbers**

In the following exercises, subtract and then check by adding.

**Translate Word Phrases to Algebraic Expressions**

In the following exercises, translate and simplify.

**Mixed Practice**

In the following exercises, simplify.

In the following exercises, translate and simplify.

Seventy-five more than thirty-five

Sixty more than ninety-three

**Subtract Whole Numbers in Applications**

In the following exercises, solve.

#### Everyday Math

#### Writing Exercises

Explain how subtraction and addition are related.

How does knowing addition facts help you to subtract numbers?

#### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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- Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
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- Book title: Prealgebra 2e
- Publication date: Mar 11, 2020
- Location: Houston, Texas
- Book URL: https://openstax.org/books/prealgebra-2e/pages/1-introduction
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**Unit Description**The student will solve contextual problems using whole numbers. All student learning outcomes for this unit must be completed without the use of a calculator. Emphasis should be placed on applications throughout the unit.

**Broad Learning Outcomes**Upon completion of Unit 0 students will be able to:

0.1 Recognize place value and names for numbers

0.2 Perform operations with whole numbers

0.3 Round whole numbers and estimation with whole numbers

0.4 Solve application problems by adding, subtracting, multiplying, or dividing whole numbers**Specific Objectives**Upon completion of Unit 0 students will be able to:

**0.1 Recognize place value and names for numbers****0.2 Perform operations with whole numbers****0.3 Round whole numbers and estimate sums, differences, products and quotients with whole numbers**

## Respond to this Question

1. Write the fraction or mixed number as a decimal. 7/8 0.78 0.875* 0.8777 1.875 2. Write the fraction or mixed number as a decimal 1 1/2 0.50 1.5* 1.05 1.1 3. Write the fraction or mixed number as a decimal. 4 3/8 0.375 4.35

A cook needs 5 cups of vegetable stock for a soup recipe. How much is this in pints? Write your answer as a whole number or a mixed number in simplest form. Include the correct unit in your answer. And the units are, fl oz, C, pt,

Draw a model for each mixed number then write the mixed number as an improper fraction 1.4 1/3=13/13 2.3 3/8=27/8 3.2 2/5=12/5 If those answers are correct, how do I draw a model?

### Spanish

Lucas's party is tomorrow. Complete the text below, by choosing and correctly conjugating the correct verb in the box. Verbs: traer, dar,salir,decir,and venir. 1. Lucas y su familia------------ una fiesta esta tarde. 2.

Write a mixed number that is equivalent to 16/3

### Algebra

a student's work to solve 2 6/7 divided by 2/7 is shown below. which of the following statements do you agree with? 2 6/7 divided by 2/7 = 19/7 divided by 2/7= 7/19 times 2/7= 7/19 times 2/7=2/19 cross out the 7 in the first one

Write 116 2/3 as a fraction or mixed number.

Write each of the fractions below as a division expression. Then write each fraction as a whole number or mixed number (I need help with this, because it confuses me as to how to write it as a division expression. Also sorry for

write each mixed number as an improper fraction 1 7/8

Write as fraction or mixed number 1.004

Write each of the following as a common fraction or mixed number. Write your answer in lowest terms and the number is 15.35? Please help

### Math againn (converting decimals)

Write 6.4% as a decimal and as a fraction. Write 2 and one fourth as a decimal and as a percent. To convert from percent to decimal simpoy divide by 100. 6.4% = 0.064. To convert to fraction (do you want a mixed fraction or not?

## How to Divide Fractions?

We know that division is a method of sharing equally and putting into equal groups. We divide a whole number by the divisor to get the quotient. Now, when we divide a fraction by another fraction, it is the same as multiplying the fraction by the reciprocal of the second fraction. The reciprocal of a fraction is a simple way of interchanging the fraction's numerator and denominator. Observe the following figure to learn a simple way of dividing fractions.

## 2.3 Models and Applications

Josh is hoping to get an A in his college algebra class. He has scores of 75, 82, 95, 91, and 94 on his first five tests. Only the final exam remains, and the maximum of points that can be earned is 100. Is it possible for Josh to end the course with an A? A simple linear equation will give Josh his answer.

Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus a charge per minute of talk-time it costs a widget manufacturer a certain amount to produce

*x*widgets per month plus monthly operating charges a car rental company charges a daily fee plus an amount per mile driven. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems.### Setting up a Linear Equation to Solve a Real-World Application

To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. Let us use the car rental example above. In this case, a known cost, such as .10/mi, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write 0.10 x . 0.10 x . This expression represents a variable cost because it changes according to the number of miles driven.

If a quantity is independent of a variable, we usually just add or subtract it, according to the problem. As these amounts do not change, we call them fixed costs. Consider a car rental agency that charges .10/mi plus a daily fee of $50. We can use these quantities to model an equation that can be used to find the daily car rental cost C . C .

When dealing with real-world applications, there are certain expressions that we can translate directly into math. Table 1 lists some common verbal expressions and their equivalent mathematical expressions.

### How To

**Given a real-world problem, model a linear equation to fit it.**- Identify known quantities.
- Assign a variable to represent the unknown quantity.
- If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.
- Write an equation interpreting the words as mathematical operations.
- Solve the equation. Be sure the solution can be explained in words, including the units of measure.

### Example 1

#### Modeling a Linear Equation to Solve an Unknown Number Problem

Find a linear equation to solve for the following unknown quantities: One number exceeds another number by 17 17 and their sum is 31. 31. Find the two numbers.

Try to

**exactly divide**(only whole number answers) both the top and bottom of the fraction by 2, 3, 5, 7 . etc, until we can't go any further.### Example: Simplify the fraction 24 108 :

÷ 2 ÷ 2 ÷ 3 *24***108**=*12***54**=*6***27**=*2***9**÷ 2 ÷ 2 ÷ 3 That is as far as we can go. The fraction simplifies to

*2***9**### Example: Simplify the fraction 10 35 :

Dividing by

**2**doesn't work because**35 can't be exactly divided by 2**(35/2 = 17½)Likewise we can't divide exactly by

**3**(10/3 = 3*1***3**and also 35/3=11*2***3**)No need to check

**4**(we checked 2 already, and 4 is just 2×2).÷ 5 *10***35**=*2***7**÷ 5 That is as far as we can go. The fraction simplifies to

*2***7**Notice that after checking 2 we didn't need to check 4 (4 is 2×2)?

We also don't need to check 6 when we have checked 2 and 3 (6 is 2x3).

In fact, when checking from smallest to largest we use prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, .

## Online Calculators and Tools by Visual Fractions

Visual Fractions started way back in 1999 as a way to help students learn about fractions and to understand them using interactive visual tools. Since then, we have expanded to become an online reference - covering fraction and math calculators, percentages, unit conversions, and more.

The main areas of the site can be explored below and we are adding more every week so please bookmark and check back later for new tools.

If you use any of the calculations, images, or snippets of content from Visual Fractions in your research or on your own website, please make sure to reference us as the source material and provide a link from your website. This helps us to keep the entire site free to use. There is a citation tool at the end of every single page on this site so you can quickly copy and paste.

Learn how to solve these kinds of problems.

Welcome to this free lesson guide where you will learn and easy two-step process for multiplying fractions by whole numbers AND multiplying whole numbers by fractions.

This complete guide to multiplying fractions by whole numbers includes several examples, an animated video mini-lesson, and a free worksheet and answer key.