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Product Estimation with Whole Numbers and Fractions

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Product Estimation with Whole Numbers and Fractions
Credit: Andrew Roberts
Source: https://www.flickr.com/photos/andrewr/5193121034/

Johnny is thinking about how much he sleeps. He estimates that a person spends about a third of the day asleep. About how many days does a a person spend asleep over the course of one year?

In this concept, you will learn how to estimate the products of whole numbers and fractions.

Estimating Products with Whole Numbers and Fractions

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

Round the whole number to the closest compatible number to the denominator. Compatible numbers are numbers that are close in value to the real number that would make it easier to find an estimate calculation.

Here is a multiplication problem.

395×103\begin{align*}395 \times 103 \end{align*}

You can find an estimate of the product by rounding the numbers to the nearest hundreds place. Compatible numbers for 395 and 103 are 400 and 100.

400×100=40,000\begin{align*}400 \times 100 = 40,000\end{align*}

To estimate the product of a whole number and a fraction, round the whole number to a number compatible to the denominator of the fraction or round the fraction a fraction compatible to the whole number.

Here is a multiplication problem involving a fraction and whole number.

39×11=\begin{align*}\frac{3}{9} \times 11= \underline{\;\;\;\;\;\;}\end{align*}

First, think about the fraction three-ninths. Three-ninths simplifies to one-third.

13×11=\begin{align*}\frac{1}{3}\times 11= \underline{\;\;\;\;\;\;}\end{align*}

Note that multiplying by one-third is also the same as dividing by 3. You can round the whole number to a number that is compatible to the denominator of the fraction, 3. A compatible whole number would be a multiple of 3 that is close to 11. 11 is close to 9 and 12. 12 is the closer compatible number.

13×12=\begin{align*}\frac{1}{3} \times 12 = \underline{\;\;\;\;\;\;}\end{align*}

Now, multiply to find an estimate product.

13×12=4\begin{align*}\frac{1}{3}\times 12=4\end{align*}

An estimate product is 4.

Here is another problem.

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

First, round the fraction to a fraction that is easy to multiply with 20. Four-sixteenths is close to five-sixteenths and simplifies to one-fourth. 20 is divisible by four. Round five-sixteenths to one-fourth.

51641614×20=14\begin{align*}\frac {5}{16} \approx \frac{4}{16} &= \frac{1}{4}\\ \frac{1}{4} \times 20 \end{align*}

Then, multiply to find an estimate product.

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

An estimate product is 5.

Examples

Example 1

Earlier, you were given a problem about Johnny's sleep.

Johnny wants to know about how many days a person spends asleep in a year. There are 365 days in a year. To find an estimate, multiply 365 by one-third.

365×13\begin{align*}365 \times \frac{1}{3}\end{align*}

First, round the whole number to a number compatible with 3. Remember that the divisibility rules for 3 states that a number is divisible by 3 if the sum of all the digits is also divisible by 3. Round 365 to 366.

365366\begin{align*}365 \approx 366\end{align*}

366×13\begin{align*}366 \times \frac{1}{3}\end{align*}

Then, multiply to find the product. Multiply by one-third is the same as dividing by 3.

366×13=122\begin{align*}366 \times \frac{1}{3}=122\end{align*}

A person spends about 122 days asleep in a year.

Example 2

Estimate the following product.

12×281\begin{align*}\frac{1}{2} \times 281\end{align*}

First, round the whole number or the fraction. You can round 281 to a number compatible to the denominator 2.

281280\begin{align*}281 \approx 280\end{align*}

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

Then, multiply 280 by one-half.

12×280=140\begin{align*}\frac{1}{2} \times 280 = 140\end{align*}

An estimate product is 140.

Example 3

Estimate the product: 10×38=\begin{align*}10\times \frac{3}{8} = \underline{\;\;\;\;\;\;\;}\end{align*}.

First, look at the fraction. Round three-eighths to a compatible number.

3848=12\begin{align*}\frac{3}{8}\approx \frac{4}{8}=\frac{1}{2}\end{align*}    10×12\begin{align*}10 \times \frac{1}{2}\end{align*}

Then, multiply to find the product.

10×12=5\begin{align*}10 \times \frac{1}{2}= 5\end{align*}

An estimate product is 5.

Example 4

Estimate the product: 12×19=\begin{align*}\frac{1}{2} \times 19= \underline{\;\;\;\;\;\;\;}\end{align*}.

First, round the whole number to a number compatible to the denominator, 2.

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

Then, multiply to find the product.

12×18=12×181=11×91=9\begin{align*}\frac{1}{2} \times 18 =\frac{1}{\cancel{2}} \times \frac{\cancel{18}}{1} = \frac{1}{1} \times \frac{9}{1}=9\end{align*}

An estimate product is 9.

Example 5

Estimate the product: 34×78=\begin{align*}\frac{3}{4} \times 78 = \underline{\;\;\;\;\;\;\;}\end{align*}.

First, round the whole number to a number compatible to the denominator, 4.

34×80\begin{align*}\frac{3}{4} \times 80 \end{align*}

Then, multiply to find the product.

34×80=34×801=31×201=60\begin{align*}\frac{3}{4} \times 80 = \frac{3}{\cancel{4}} \times \frac{\cancel{80}}{1}= \frac{3}{1} \times \frac{20}{1}= 60\end{align*}

An estimate product is 60.

Review

Estimate the products.

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*}

To see the Review answers, open this PDF file and look for section 7.2.

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Color Highlighted Text Notes

Vocabulary Language: English

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.