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7.3: Product Estimation with Whole Numbers and Fractions

Difficulty Level: At Grade Created by: CK-12
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Practice Product Estimation with Whole Numbers and Fractions

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Remember Julie and the rainforest?

Well, Julie calculated that the average amount of rainfall in the rainforest per day was about 18\begin{align*} \frac{1}{8}\end{align*} of an inch. If Julie wanted to figure out how much rainfall there would be for one year, how could she do this?

There are 365 days in a year. If Julie multiplied 365 by one - eighth she could figure it out.

Or she could estimate. Julie could estimate the following product.

365×18\begin{align*}365 \times \frac{1}{8}\end{align*}

How could she do this?

This Concept will teach you how to estimate the products of whole numbers and fractions. By the end of the Concept, you will understand how to help Julie figure this out.

Guidance

We can estimate products of whole numbers and fractions. When we estimate, we are looking for an answer that is reasonable but need not be exact.

Before we look at how to do it, we need to know that the commutative property applies to multiplying fractions and whole numbers. It doesn’t matter which order you multiply in, the answer will be the same.

6×12=12×6\begin{align*}6 \times \frac{1}{2} = \frac{1}{2} \times 6\end{align*}

It doesn’t matter which order we write the numbers in, the answer will remain the same. This is an illustration of the commutative property.

How can we estimate the product of a whole number and a fraction?

To estimate the product, we have to use some reasoning skills.

39×12=\begin{align*}\frac{3}{9} \times 12 = \underline{\;\;\;\;\;\;\;}\end{align*}

To work on this problem, we have to think about three-ninths. Three-ninths simplifies to one-third. Now we can find one-third of 12. Multiplying by one-third is the same as dividing by three.

516×20=\begin{align*}\frac{5}{16} \times 20 = \underline{\;\;\;\;\;\;\;}\end{align*}

To estimate this problem, we must think about a fraction that is easy to divide into twenty, but that is close to five-sixteenths. Four-sixteenths is close to five-sixteenths and it simplifies to one-fourth. Twenty is divisible by four, so we can rewrite the problem and solve.

41614×20=14=5\begin{align*}\frac{4}{16} & = \frac{1}{4}\\ \frac{1}{4} \times 20 & = 5\end{align*}

Remember that multiplying by one-fourth is the same as dividing by four, so our answer is five.

Our estimate is five.

Practice a few of these on your own. Estimate these products.

Example A

8×38=\begin{align*}8 \times \frac{3}{8} = \underline{\;\;\;\;\;\;\;}\end{align*}

Solution:3\begin{align*}3\end{align*}

Example B

12×18=\begin{align*}\frac{1}{2} \times 18 = \underline{\;\;\;\;\;\;\;}\end{align*}

Solution:9\begin{align*}9\end{align*}

Example C

34×75=\begin{align*}\frac{3}{4} \times 75 = \underline{\;\;\;\;\;\;\;}\end{align*}

Solution:25\begin{align*}25\end{align*}

Now back to Julie and the rainforest. Here is the original problem once again.

Well, Julie calculated that the average amount of rainfall in the rainforest per day was about 18\begin{align*} \frac{1}{8}\end{align*} of an inch. If Julie wanted to figure out how much rainfall there would be for one year, how could she do this?

There are 365 days in a year. If Julie multiplied 365 by one - eighth she could figure it out.

Or she could estimate. Julie could estimate the following product.

365×18\begin{align*}365 \times \frac{1}{8}\end{align*}

How could she do this?

To do this, we could first round 365 up to 400.

400 is easily divisible by 8.

There is an average of 50 inches of rain in the rainforest per year.

Vocabulary

Here are the vocabulary words in this Concept.

Multiplication
“of”
means multiply in a word problem
Product
the answer to a multiplication problem
Estimate
to find a reasonable answer that is not exact but is close to the actual answer.

Guided Practice

Here is one for you to try on your own.

Estimate the following product.

12×280\begin{align*}\frac{1}{2} \times 280\end{align*}

To figure out this estimate, let's first round 280\begin{align*}280\end{align*}.

280\begin{align*}280\end{align*} rounds up to 300\begin{align*}300\end{align*}.

Now we can easily find half of 300.

Our answer is 150\begin{align*}150\end{align*}.

Video Review

Here is a video for review.

Multiplying Fractions and Whole Numbers- This video involves skills needed in this Concept.

Practice

Directions: Estimate the products of the following fractions and whole numbers.

1. 6×12=\begin{align*}6 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

2. 16×12=\begin{align*}16 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

3. 26×12=\begin{align*}26 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

4. 36×13=\begin{align*}36 \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

5. 40×15=\begin{align*}40 \times \frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

6. 20×14=\begin{align*}20 \times \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

7. 30×12=\begin{align*}30 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

8. 100×110=\begin{align*}100 \times \frac{1}{10} = \underline{\;\;\;\;\;\;\;}\end{align*}

9. 60×13=\begin{align*}60 \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

10. 90×13=\begin{align*}90 \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

11. 33×111=\begin{align*}33 \times \frac{1}{11} = \underline{\;\;\;\;\;\;\;}\end{align*}

12. 44×14=\begin{align*}44 \times \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

13. 36×112=\begin{align*}36 \times \frac{1}{12} = \underline{\;\;\;\;\;\;\;}\end{align*}

14. 50×125=\begin{align*}50 \times \frac{1}{25} = \underline{\;\;\;\;\;\;\;}\end{align*}

15. 75×34=\begin{align*}75 \times \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

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Color Highlighted Text Notes

Vocabulary Language: English

TermDefinition
Estimate To estimate is to find an approximate answer that is reasonable or makes sense given the problem.
multiplication Multiplication is a simplified form of repeated addition. Multiplication is used to determine the result of adding a term to itself a specified number of times.
Product The product is the result after two amounts have been multiplied.

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Date Created:
Oct 29, 2012
Sep 04, 2016
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MAT.ARI.335.4.L.1