Whole Numbers: Learn It 5

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Whole Numbers: Learn It 5

Subtracting 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 [latex]3[/latex] from [latex]7[/latex] is:

[latex]7 - 3[/latex]

We read [latex]7 - 3[/latex] as seven minus three and the result is the difference of seven and three.

subtraction notation

To describe subtraction, we can use symbols and words.

Operation	Subtraction
Notation	[latex]-[/latex]
Expression	[latex]7 - 3[/latex]
Read as	seven minus three
Result	the difference of [latex]7[/latex] and [latex]3[/latex]

You may see other words in problems besides “difference” that indicate the need for subtraction. Below are other subtraction words you might come across.

Operation	Word Phrase	Example	Expression
Subtraction
	minus	[latex]5[/latex] minus [latex]1[/latex]	[latex]5 - 1[/latex]
	difference	the difference of [latex]9[/latex] and [latex]4[/latex]	[latex]9 - 4[/latex]
	decreased by	[latex]7[/latex] decreased by [latex]3[/latex]	[latex]7 - 3[/latex]
	less than	[latex]5[/latex] less than [latex]8[/latex]	[latex]8 - 5[/latex]
	subtracted from	[latex]1[/latex] subtracted from [latex]6[/latex]	[latex]6 - 1[/latex]

Translate from math notation to words:

[latex]8 - 1[/latex]
[latex]26 - 14[/latex]

Addition and subtraction are inverse operations. Addition undoes subtraction, and subtraction undoes addition.
We know [latex]7 - 3=4[/latex] because [latex]4+3=7[/latex]. Knowing all the addition number facts will help with subtraction. Then we can check subtraction by adding.

[latex]7-3=4[/latex]	because	[latex]4+3=7[/latex]
[latex]13-8=5[/latex]	because	[latex]5+8=13[/latex]
[latex]43-26=17[/latex]	because	[latex]17+26=43[/latex]

Subtract and then check by adding:

[latex]9 - 7[/latex]
[latex]8 - 3[/latex]

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.

Subtract the follwoing. Check your answers by adding.

[latex]89 - 61[/latex]

What happens if you need to subtract a larger number from a smaller one? In the process of subtracting larger numbers from smaller ones in mathematics, the concept of “borrowing” plays a crucial role. Let’s use our base – [latex]10[/latex] model to find out.

The graphic below shows the subtraction of [latex]43-26[/latex].

Because [latex]43 - 26[/latex] means [latex]43[/latex] take away [latex]26[/latex], we begin by modeling the [latex]43[/latex].

An image containing two items. The first item is 4 horizontal rods containing 10 blocks each. The second item is 3 individual blocks.

Now, we need to take away [latex]26[/latex], which is [latex]2[/latex] tens and [latex]6[/latex] ones. We cannot take away [latex]6[/latex] ones from [latex]3[/latex] ones. So, we exchange [latex]1[/latex] ten for [latex]10[/latex] ones.

This figure contains two groups. The first group on the left includes 3 rows of blue base 10 blocks and 1 red row of 10 blocks. This is labeled 4 tens. Alongside the first row of ten blocks are 3 individual blocks. This is labeled 3 ones. An arrow points to the right to the second group in which there are three rows of 10 base blocks labeled 3 tens. Next to this is a row of 3 blue individual blocks and two rows each with five individual blocks in red. This is labeled 13 ones.

Now we can take away [latex]2[/latex] tens and [latex]6[/latex] ones.

This image includes one row of base ten blocks at the top of the image; Next to it are seven individual blocks. Below this, is a group of two rows of base ten blocks, and two rows of 3 individual blocks with a circle around all. The arrow points to the right and shows one row of ten blocks and seven individual blocks underneath.

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

[latex]43 - 26=17[/latex]

When we do this without the model, we say we borrow [latex]1[/latex] from the tens place and add [latex]10[/latex] to the ones place.

Subtracting numbers might appear challenging, but it can be straightforward and engaging with the right problem-solving strategies! When subtracting one number from another, try using the steps below.

Write the numbers so each place value lines up vertically.
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.
Continue subtracting each place value from right to left, borrowing if needed.
Check by adding.

Subtract the following and then check by adding:

[latex]207 - 64[/latex]
[latex]910 - 586[/latex]
[latex]2,162 - 479[/latex]

Write the numbers so each place value lines up vertically.
Subtract the ones. [latex]7 - 4=3[/latex]. Write the [latex]3[/latex] in the ones place in the difference. Write the [latex]3[/latex] in the ones place in the difference.
Subtract the tens. We cannot subtract [latex]6[/latex] from [latex]0[/latex] so we borrow [latex]1[/latex] hundred and add [latex]10[/latex] tens to the [latex]0[/latex] tens we had. This makes a total of [latex]10[/latex] tens. We write [latex]10[/latex] above the tens place and cross out the [latex]0[/latex]. Then we cross out the [latex]2[/latex] in the hundreds place and write [latex]1[/latex] above it.
Now we subtract the tens. [latex]10 - 6=4[/latex]. We write the 4 in the tens place in the difference.
Finally, subtract the hundreds. There is no digit in the hundreds place in the bottom number so we can imagine a [latex]0[/latex] in that place. Since [latex]1 - 0=1[/latex], we write [latex]1[/latex] in the hundreds place in the difference.
Check by adding. Our answer is correct.

Write the numbers so each place value lines up vertically.
Subtract the ones. We cannot subtract [latex]6[/latex] from [latex]0[/latex], so we borrow [latex]1[/latex] ten and add [latex]10[/latex] ones to the [latex]10[/latex] ones we had. This makes [latex]10[/latex] ones. We write a [latex]0[/latex] above the tens place and cross out the [latex]1[/latex]. We write the [latex]10[/latex] above the ones place and cross out the [latex]0[/latex]. Now we can subtract the ones. [latex]10 - 6=4[/latex].
Write the 4 in the ones place of the difference.
Subtract the tens. We cannot subtract [latex]8[/latex] from [latex]0[/latex], so we borrow [latex]1[/latex] hundred and add [latex]10[/latex] tens to the [latex]0[/latex] tens we had, which gives us [latex]10[/latex] tens. Write [latex]8[/latex] above the hundreds place and cross out the [latex]9[/latex]. Write [latex]10[/latex] above the tens place.
Now we can subtract the tens. [latex]10 - 8=2[/latex] .
Subtract the hundreds place. [latex]8 - 5=3[/latex] Write the [latex]3[/latex] in the hundreds place in the difference.
Check by adding. Our answer is correct.

Write the numbers so each place values line up vertically.
Subtract the ones. Since we cannot subtract [latex]9[/latex] from [latex]2[/latex], borrow [latex]1[/latex] ten and add [latex]10[/latex] ones to the [latex]2[/latex] ones to make [latex]12[/latex] ones. Write [latex]5[/latex] above the tens place and cross out the [latex]6[/latex]. Write [latex]12[/latex] above the ones place and cross out the [latex]2[/latex].
Now we can subtract the ones.	[latex]12 - 9=3[/latex]
Write [latex]3[/latex] in the ones place in the difference.
Subtract the tens. Since we cannot subtract [latex]7[/latex] from [latex]5[/latex], borrow [latex]1[/latex] hundred and add [latex]10[/latex] tens to the [latex]5[/latex] tens to make [latex]15[/latex] tens. Write [latex]0[/latex] above the hundreds place and cross out the [latex]1[/latex]. Write [latex]15[/latex] above the tens place.
Now we can subtract the tens.	[latex]15 - 7=8[/latex]
Write [latex]8[/latex] in the tens place in the difference.
Now we can subtract the hundreds.
Write [latex]6[/latex] 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 [latex]0[/latex]. [latex]1 - 0=1[/latex]. Write [latex]1[/latex] in the thousands place of the difference.
Check by adding. [latex]\begin{array}{}\\ \stackrel{1}{1},\stackrel{1}{6}\stackrel{1}{8}3\hfill \\ \underset{\text{______}}{+479}\hfill \\ 2,162\quad\checkmark \hfill \end{array}\quad\checkmark[/latex] Our answer is 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.

The temperature in Chicago one morning was [latex]73[/latex] degrees Fahrenheit. A cold front arrived and by noon the temperature was [latex]27[/latex] degrees Fahrenheit. What was the difference between the temperature in the morning and the temperature at noon?

Write a phrase.	the difference of [latex]73[/latex] and [latex]27[/latex]
Translate to math notation. Difference tells us to subtract.	[latex]73 - 27[/latex]
Then we do the subtraction.
Write a sentence to answer the question.	The difference in temperatures was [latex]46[/latex] degrees Fahrenheit.