Introduction

Subtracting large numbers without using a calculator takes some effort. However, in some circumstances, it is sufficient to find an estimate for a particular difference. Computing an estimate is usually faster and easier than solving the original problem.

To estimate the answer to a subtraction problem, we round the numbers first and then subtract them.

To illustrate this, let's find an estimate for the value of

481 - 73.

Let's start by rounding both numbers to the nearest ten:

\begin{array}{ccccl} 481 & - & 73 \\ \big\downarrow && \big\downarrow \\ \color{blue}480 & - & \color{blue}70 \end{array}

Next, we calculate the difference {\color{blue}{480}} - {\color{blue}{70}}\mathbin{:}

\require{cancel} \begin{array}{cccccccc} & & \!\!\! 4 \!\!\!& \!\!\! 8 \!\!\!& \!\!\! 0 \!\!\! \\ \!\!\!\!-\!\!\!\! & & & \!\!\! 7 \!\!\!& \!\!\! 0 \!\!\! \\ \hline & & \!\!\! 4 \!\!\!& \!\!\! 1 \!\!\!& \!\!\! 0 \!\!\! \end{array}

Therefore,

481 - 73 \approx 410.

The symbol \approx means "is approximately equal to."

The estimate we've obtained is pretty close to the actual value of the difference. In fact, 481 - 73=408.

In this example, we rounded to the nearest ten. In the next example, we'll estimate a difference by rounding both numbers to the nearest hundred.

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Example: Estimating Differences Containing Numbers Up to Three Digits

Sandy bought 432 grams of chocolate chips and used 302 grams to make cookies. By rounding each value to the nearest hundred, estimate the weight of the remaining chocolate chips.

EXPLANATION

To estimate the weight of the remaining chocolate chips, we need to approximate the difference

432 - 302.

First, we round both numbers to the nearest hundred:

\begin{array}{ccccl} 432 & - & 302\\ \big\downarrow && \big\downarrow \\ \color{blue}400 & - & \color{blue}300 \end{array}

Next, we calculate the difference {\color{blue}{400}} - {\color{blue}{300}}\mathbin{:}

\require{cancel} \begin{array}{cccccccc} & & \!\!\! 4 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\! \\ \!\!\!\!-\!\!\!\! & & \!\!\! 3 \!\!\! & \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\! \\ \hline & & \!\!\! 1 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\! \end{array}

So, we have

432 - 302 \approx 100.

Therefore, Sandy has approximately 100 grams of chocolate chips remaining.

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Practice: Estimating Differences Containing Numbers Up to Three Digits

Rounding each number to the nearest hundred, the difference $715 - 397$ is approximately

a
b
c
d
e
Practice: Estimating Differences Containing Numbers Up to Three Digits

Rounding each number to the nearest ten, the difference $802 - 157$ is approximately

a
b
c
d
e
Practice: Estimating Differences Containing Numbers Up to Three Digits

Rachel had $\$491$ in her wallet. Then, she bought some goods for $\$114.$ By rounding each value to the nearest hundred, estimate the amount of money Rachel has after purchasing the goods.

a
 $\$200$
b
 $\$500$
c
 $\$400$
d
 $\$300$
e
 $\$600$
Example: Estimating Differences Containing Numbers Up to Five Digits

Estimate the difference 83,250 - 15,845 by rounding each number to the nearest thousand.

EXPLANATION

First, we round both numbers to the nearest thousand:

\begin{array}{ccccl} 83,250 & - & 15,845\\ \big\downarrow && \big\downarrow \\ \color{blue}83,000 & - & \color{blue}16,000 \end{array}

Next, we calculate the difference {\color{blue}{83,000}} - {\color{blue}{16,000}}\mathbin{:}

\require{cancel} \begin{array}{cccccccc} & & \!\!\! \color{blue}\substack{ \\ 7 } & \!\!\! \color{blue}\substack{ \\ 13 } & \!\!\! & \!\!\! & \!\!\! \\ & & \!\!\! \cancel{8} \!\!\!& \!\!\! \cancel{3} \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\! \\ \!\!\!\!-\!\!\!\! & & \!\!\! 1 \!\!\!& \!\!\! 6 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\! \\ \hline & & \!\!\! 6 \!\!\!& \!\!\! 7 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\! \end{array}

Therefore, 83,250 - 15,845 is approximately 67,000.

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Practice: Estimating Differences Containing Numbers Up to Five Digits

Estimate the difference $47,192 - 11,940$ by rounding each number to the nearest ten thousand.

a
 $35,000$
b
 $40,000$
c
 $30,000$
d
 $25,000$
e
 $20,000$
Practice: Estimating Differences Containing Numbers Up to Five Digits

Rounding each number to the nearest thousand, the difference $24,544 - 6,368$ is approximately

a
b
c
d
e
Practice: Estimating Differences Containing Numbers Up to Five Digits

In a lottery game, Chelsea won $\$57,815.$ After taxes, her profit was $\$43,939.$ Rounding each number to the nearest hundred, estimate how much money Chelsea had to pay in taxes.

a
b
c
d
e
Example: Estimating Differences Containing Numbers Up to Seven Digits

Estimate the difference 4,755,450 - 804,001 by rounding each number to the nearest hundred thousand.

EXPLANATION

First, we round both numbers to the nearest hundred thousand:

\begin{array}{ccccl} 4,755,450 & - & 804,001\\ \big\downarrow && \big\downarrow \\ \color{blue}4,800,000 & - & \color{blue}800,000 \end{array}

Next, we calculate the difference {\color{blue}{4,800,000}} - {\color{blue}{800,000}}\mathbin{:}

\begin{array}{cccccccc} & & \!\!\! 4 \!\!\!& \!\!\! 8 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\! \\ \!\!\!\!-\!\!\!\! & & & \!\!\! 8 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\! \\ \hline & & \!\!\! 4 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\!& \!\!\! 0 \!\!\! \end{array}

Therefore, 4,755,450 - 804,001 is approximately 4,000,000.

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Practice: Estimating Differences Containing Numbers Up to Seven Digits

Estimate the difference $8,405,750- 6,710,001$ by rounding each number to the nearest million.

a
 $1,500,000$
b
 $1,000,000$
c
 $2,000,000$
d
 $3,000,000$
e
 $1,700,000$
Practice: Estimating Differences Containing Numbers Up to Seven Digits

Rounding each number to the nearest hundred thousand, the difference $3,825,302 - 1,872,501$ is approximately

a
b
c
d
e
Practice: Estimating Differences Containing Numbers Up to Seven Digits

Estimate the difference $2,154,627 - 752,554$ by rounding each number to the nearest hundred thousand.

a
 $2,000,000$
b
 $1,900,000$
c
 $1,000,000$
d
 $1,700,000$
e
 $1,400,000$
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