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Estimation of Whole Number Multiplication and Division

Estimate products and quotients with rounding.

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Estimation of Whole Number Multiplication and Division

License: CC BY-NC 3.0

Craig placed an order for 1700 highlighters for a special event hosted by his company. However, instead of placing a single order, he accidentally subscribed to receive 1700 highlighters every month for a year. As the purchase was non-refundable, Craig needs to know how many highlighters to expect so that he can pitch to his boss other ways to use them.  He is running into a meeting and doesn’t have time to make a calculation. How can Craig estimate the number of highlighters the company will receive over the course of the year?

In this concept, you will learn how to estimate products and quotients of whole numbers using rounding.

Estimating Multiplying and Dividing Whole Numbers

Just as you can estimate sums and differences, you can also estimate products and quotients.

Remember, a product is the answer to a multiplication problem. A quotient is the answer to a division problem.To estimate products and quotients, first round according to the following rules:

1. If the number to the right of the digit you are rounding is less than 5, round down (keep the number being rounded as it is).
2. If the number to the right of the digit you are rounding is greater than 5, round up.
Let's look at this problem.

12×19=\begin{align*}12 \times 19 = \underline{\;\;\;\;\;\;\;}\end{align*}

First, round the first number to the nearest ten. Since the number in the ones place is less than 5, round down (the digit in the tens place is left as-is). 12 rounds to 10.

Next, round the second number to the nearest ten. Since the number in the ones place is a 9, which is more than 5, round the 1 from the tens place up to a 2. 19 rounds to 20.

Finally, multiply the rounded numbers by each other:

10×20=200\begin{align*} 10 \times 20 = 200\end{align*}

The estimate of the product is 200.

Here is an example of a division problem. Estimate the quotient.

32÷11\begin{align*}32 \div 11\end{align*}

First, round each number to the nearest ten. 32 rounds down to 30, since the number in the ones place is smaller than 5. 11 rounds down to 10, since the number in the ones place is smaller than 5.

Next find the quotient of the estimated values:

30÷10=3\begin{align*}30 \div 10 = 3\end{align*}

The estimated quotient is 3.

Sometimes, when working with division, you need to find a compatible number instead of a rounded number. A compatible number is one that is easily divisible.

Let’s look at estimating a problem that needs a compatible number.

2321÷8=\begin{align*}2321 \div 8 = \underline{\;\;\;\;\;\;\;}\end{align*}

Normally, you would round 2321 down to 2300, but 2300 is not easily divisible by 8. The nearest hundred easily divisible by 8 is 2400, because 24 divided by 8 is 3. This makes 2400 a compatible number.

2400÷8=300\begin{align*} 2400 \div 8 = 300\end{align*}

The estimate is 300.

Remember to keep the following things in mind when estimating:

1. The estimate must make sense for the problem.
2. The estimate must be reasonable.
3. The estimate must be close to the exact answer.
4. If none of the above fit the estimate, use exact math to find the solution.

Examples

Example 1

Earlier, you were given a problem about Craig and his non-refundable highlighters.

Craig needs to estimate the total of 1700 highlighters every month for 12 months.

First, round the numbers:

1700 rounds up to 2000

12 rounds down to 10

Next, use mental math.

2,000×10=20,000\begin{align*}2,000 \times 10 = 20,000\end{align*}

The estimation is 20,000.

Craig can report in the meeting that he has a plan to make use of approximately 20,000 highlighters this year.

Example 2

Estimate the quotient.

869÷321=\begin{align*}869 \div 321 = \underline{\;\;\;\;\;\;\;}\end{align*}

First, round each number to the nearest hundred.

869 rounds to 900321 rounds to 300\begin{align*}869 \text{ rounds to } 900\\ 321 \text{ rounds to } 300\\\end{align*}

Then, estimate.

900÷300=3\begin{align*} 900 \div 300 = 3\end{align*}

The estimated quotient is 3.

Example 3

Estimate the product.

34×18=\begin{align*}34 \times 18 =\underline{\;\;\;\;\;\;\;}\end{align*}

First round 34 to the nearest ten. Since the number in the ones place is less than 5, round down (keep the 3 in the tens place). 34 rounds to 30.

Next round 18 to the nearest ten. Since the number in the ones place is greater than 5, round up. 18 rounds to 20.

Finally, multiply the rounded values:

30×  2000+  600600\begin{align*}30\\ \underline{\times \ \ 20} &\\ 00 & \\ \underline{+ \ \ 600} & \\ 600 & \\\end{align*}

The estimated product is 600.

Example 4

Estimate the product.

187×11=\begin{align*}187 \times 11 = \underline{\;\;\;\;\;\;\;}\end{align*}

First, round 187 to the nearest hundred. Since the number in the tens place is greater than 5, round up. 187 rounds to 200.

Next, round 11 to the nearest 10. Since the number in the ones place is less than 5, round down. 11 rounds to 10.

Finally, multiply the rounded values:

200×   10000+  20002000\begin{align*}200 &\\ \underline{ \times \ \ \ 10} & \\ 000 &\\ \underline{+ \ \ 2000} & \\ 2000 & \\\end{align*}

The estimated product is 2000.

Example 5

Estimate the quotient.

120÷11=\begin{align*}120 \div 11 = \underline{\;\;\;\;\;\;\;}\end{align*}

First, note that 120 will divide evenly by 10, so just round the second number, leaving 120 as it is.

Next, since the number in the ones place of the second number is less than 5, round down. 11 rounds to 10.

Finally, use mental math to divide 120 by 10.

The estimated quotient is 12.

Review

Estimate the following products and quotients.

1. 17×12=\begin{align*}17 \times 12 = \underline{\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}22 \times 18 = \underline{\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}9 \times 18 = \underline{\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}7 \times 23 = \underline{\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}36 \times 40 = \underline{\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}13 \times 31 = \underline{\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}9 \times 27 = \underline{\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}11 \times 32 = \underline{\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}19 \times 33 = \underline{\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}22 \times 50 = \underline{\;\;\;\;\;\;\;}\end{align*}
11. \begin{align*}43 \div 6 = \underline{\;\;\;\;\;\;\;}\end{align*}
12. \begin{align*}19 \div 10 = \underline{\;\;\;\;\;\;\;}\end{align*}
13. \begin{align*}44 \div 8 = \underline{\;\;\;\;\;\;\;}\end{align*}
14. \begin{align*}72 \div 7 = \underline{\;\;\;\;\;\;\;}\end{align*}
15. \begin{align*}17 \div 8 = \underline{\;\;\;\;\;\;\;}\end{align*}
16. \begin{align*}43 \div 9 = \underline{\;\;\;\;\;\;\;}\end{align*}
17. \begin{align*}62 \div 8 = \underline{\;\;\;\;\;\;\;}\end{align*}
18. \begin{align*}102 \div 18 = \underline{\;\;\;\;\;\;\;}\end{align*}
19. \begin{align*}395 \div 11 = \underline{\;\;\;\;\;\;\;}\end{align*}
20. \begin{align*}778 \div 22 = \underline{\;\;\;\;\;\;\;}\end{align*}

To see the Review answers, open this PDF file and look for section 1.6.

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Vocabulary Language: English

Compatible number

A number that is easily divisible by the divisor in an estimation problem.

Estimate

To estimate is to find an approximate answer that is reasonable or makes sense given the problem.

Factors

Factors are numbers or values multiplied to equal a product.

Product

The product is the result after two amounts have been multiplied.

Quotient

The quotient is the result after two amounts have been divided.