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Product Estimation with Mixed Numbers/Fractions

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Remember how Michelle was working with bake sale statistics in the Products of Mixed Numbers Concept? Well, look at this dilemma.

Michelle knows that \frac{3}{4} of the cookies left were peanut butter cookies. There were 5 \frac{3}{4} of a dozen cookies left. Michelle wants to know what \frac{3}{4} of 5 \frac{3}{4} is, then she will know about how many of the left - over cookies were peanut butter.

Do you know how to do this?

Because Michelle doesn't need an exact amount, we can estimate the product.

This Concept will teach you how to estimate the products of fractions and mixed numbers.

Guidance

By now you are very familiar with estimation as a tool for getting an approximate sense of the value of numbers, sums of numbers and differences of numbers.

Now, we are going to add our knowledge of estimation to get approximate answers for products of fractions.

Just like rounding whole numbers, we can find approximate values of fractions by comparing the fractions to three benchmarks, 0, \frac{1}{2} and 1.

Is the fraction closer to 0, \frac{1}{2} or 1? If it’s closest to one-half, we say that the value of the fraction is “about \frac{1}{2} .”

We can use these approximate values of fractions to estimate the products of fractions and mixed numbers.

First, we approximate the value of each fraction or mixed number using the benchmarks 0, \frac{1}{2} and 1. Next, we find the product of the approximate values. Even when you are asked to find an exact answer, estimation is a useful way to get an idea of a reasonable solution to a problem. Once you have finished solving for an exact answer of a problem, you can check your answer against the estimate. Refer back to the following steps if necessary.

\frac{19}{20} \cdot \frac{6}{7}

First, we have to take each fraction and find its benchmark. Nineteen-twentieths is close to 1. Six-sevenths is also close to one.

1 \times 1 = 1

We can say that \frac{19}{20} \cdot \frac{6}{7} is approximately 1.

What about mixed numbers?

Mixed numbers work the same way except that your benchmarks will have whole numbers in them too. Let’s look at an example.

3 \frac{6}{8} \cdot 5 \frac{1}{10}

Three and six-eighths is closest to 4.

Five and one-tenths is closest to 5.

4 \times 5 = 20

We can say that 3 \frac{6}{8} \cdot 5 \frac{1}{10} is approximately 20.

Now it's time to try a few on your own. Estimate each product using benchmarks.

Example A

\frac{6}{12} \cdot \frac{10}{11}

Solution: \frac{1}{2}

Example B

\frac{9}{10} \cdot \frac{5}{6}

Solution: 1

Example C

4 \frac{1}{8} \cdot 2 \frac{11}{13}

Solution: 12

Here is the original problem once again.

Michelle knows that \frac{3}{4} of the cookies left were peanut butter cookies. There were 5 \frac{3}{4} of a dozen cookies left. Michelle wants to know what \frac{3}{4} of 5 \frac{3}{4} is, then she will know about how many of the left - over cookies were peanut butter.

Do you know how to do this?

Because Michelle doesn't need an exact amount, we can estimate the product.

To begin, we have to use benchmarks for the dozens left in the problem. We need to find three - fourths of that number, so we don't use a benchmark for three - fourths. If we did, we would end up with a value larger than the number of dozens left over.

5 \frac{3}{4} = 6

Now we multiply.

6 \times \frac{3}{4} = 4 \frac{1}{2} dozens left were peanut butter.

This is our answer.

Vocabulary

Fraction
a part of a whole.
Mixed Number
a whole number and a fraction
Product
the answer in a multiplication problem
Estimate
an approximate solution

Guided Practice

Here is one for you to try on your own.

4 \frac{5}{6} \times 2 \frac{1}{9}

Answer

First, let's use a benchmark for each mixed number.

 4 \frac{5}{6} = 5

2 \frac{1}{9} = 2

5 \times 2 = 10

Our answer is 10.

Video Review

This is a James Sousa video on multiplying fractions. It is a supporting video to this Concept.

Practice

Directions: Estimate the product.

1. \frac{1}{9} \cdot \frac{4}{5}

2. 12 \cdot \frac{6}{7}

3. \frac{18}{37} \cdot \frac{10}{11}

4. \frac{13}{15} \cdot \frac{4}{9}

5. 6 \frac{2}{3} \cdot 2 \frac{6}{11}

6. 5 \frac{27}{29} \cdot 3 \frac{1}{18}

7. 4 \frac{6}{7} \cdot 1 \frac{4}{7}

8. 4 \frac{15}{16} \cdot 7 \frac{2}{21}

9. 4 \frac{15}{16} \cdot 7 \frac{2}{21}

10. 3 \frac{5}{6} \cdot 1 \frac{1}{21}

11. 6 \frac{15}{16} \cdot 7 \frac{1}{9}

12. 5 \frac{11}{12} \cdot 3 \frac{12}{80}

13. 4 \frac{18}{60} \cdot 7 \frac{1}{21}

14. 13 \frac{4}{5} \cdot 6 \frac{1}{10}

15. 15 \frac{11}{13} \cdot 11\frac{9}{10}

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