8.4: Estimation by Rounding Fractions

Learning Objectives

• be able to estimate the sum of two or more fractions using the technique of rounding fractions

Estimation by rounding fractions is a useful technique for estimating the result of a computation involving fractions. Fractions are commonly rounded to \(\dfrac{1}{4}\), \(\dfrac{1}{2}\), \(\dfrac{3}{4}\), 0, and 1. Remember that rounding may cause estimates to vary.

Sample Set A

Make each estimate remembering that results may vary.

Estimate \(\dfrac{3}{5} + \dfrac{5}{12}\).

Solution

Notice that \(\dfrac{3}{5}\) is about \(\dfrac{1}{2}\), and that \(\dfrac{5}{12}\) is about \(\dfrac{1}{2}\).

Thus, \(\dfrac{3}{5} + \dfrac{5}{12}\) is about \(\dfrac{1}{2} + \dfrac{1}{2} = 1\). In fact, \(\dfrac{3}{5} + \dfrac{5}{12} = \dfrac{61}{60}\), a little more than 1.

Sample Set A

Estimate \(5 \dfrac{3}{8} + 4 \dfrac{9}{10} + 11 \dfrac{1}{5}\).

Solution

Adding the whole number parts, we get 20. Notice that \(\dfrac{3}{8}\) is close to \(\dfrac{1}{4}\), \(\dfrac{9}{10}\) is close to 1, and \(\dfrac{1}{5}\) is close to \(\dfrac{1}{4}\). Then \(\dfrac{3}{8} + \dfrac{9}{10} + \dfrac{1}{5}\) is close to \(\dfrac{1}{4} + 1 + \dfrac{1}{4} = 1 \dfrac{1}{2}\).

Thus, \(5 \dfrac{3}{8} + 4 \dfrac{9}{10} + 11 \dfrac{1}{5}\) is close to \(20 + 1 \dfrac{1}{2} = 21 \dfrac{1}{2}\).

In fact, \(5 \dfrac{3}{8} + 4 \dfrac{9}{10} + 11 \dfrac{1}{5} = 21 \dfrac{19}{40}\), a little less than \(21 \dfrac{1}{2}\).

Practice Set A

Use the method of rounding fractions to estimate the result of each computation. Results may vary.

\(\dfrac{5}{8} + \dfrac{5}{12}\)

Answer

Results may vary. \(\dfrac{1}{2} + \dfrac{1}{2} = 1\). In fact, \(\dfrac{5}{8} + \dfrac{5}{12} = \dfrac{25}{24} = 1 \dfrac{1}{24}\)

Practice Set A

\(\dfrac{7}{9} + \dfrac{3}{5}\)

Answer

Results may vary. \(1 + \dfrac{1}{2} = 1 \dfrac{1}{2}\). In fact, \(\dfrac{7}{9} + \dfrac{3}{5} = 1 \dfrac{17}{45}\)

Practice Set A

\(8 \dfrac{4}{15} + 3 \dfrac{7}{10}\)

Answer

Results may vary. \(8 \dfrac{1}{4} + 3 \dfrac{3}{4} = 11 + 1 = 12\). In fact, \(8 \dfrac{4}{15} + 3 \dfrac{7}{10} = 11 \dfrac{29}{30}\)

Practice Set A

\(16 \dfrac{1}{20} + 4 \dfrac{7}{8}\)

Answer

Results may vary. \((16 + 0) + (4 + 1) = 16 + 5 = 21\). In fact, \(16 \dfrac{1}{20} + 4 \dfrac{7}{8} = 20 \dfrac{37}{40}\)

Exercises

Estimate each sum or difference using the method of rounding. After you have made an estimate, find the exact value of the sum or difference and compare this result to the estimated value. Result may vary.

Exercise \(\PageIndex{1}\)

\(\dfrac{5}{6} + \dfrac{7}{8}\)

Answer

\(1 + 1 = 2(1 \dfrac{17}{24})\)

Exercise \(\PageIndex{2}\)

\(\dfrac{3}{8} + \dfrac{11}{12}\)

Exercise \(\PageIndex{3}\)

\(\dfrac{9}{10} + \dfrac{3}{5}\)

Answer

\(1 + \dfrac{1}{2} = 1 \dfrac{1}{2} (1 \dfrac{1}{2})\)

Exercise \(\PageIndex{4}\)

\(\dfrac{13}{15} + \dfrac{1}{20}\)

Exercise \(\PageIndex{5}\)

\(\dfrac{3}{20} + \dfrac{6}{25}\)

Answer

\(\dfrac{1}{4} + \dfrac{1}{4} = \dfrac{1}{2} (\dfrac{39}{100})\)

Exercise \(\PageIndex{6}\)

\(\dfrac{1}{12} + \dfrac{4}{5}\)

Exercise \(\PageIndex{7}\)

\(\dfrac{15}{16} + \dfrac{1}{12}\)

Answer

\(1 + 0 = 1 (1 \dfrac{1}{48})\)

Exercise \(\PageIndex{8}\)

\(\dfrac{29}{30} + \dfrac{11}{20}\)

Exercise \(\PageIndex{9}\)

\(\dfrac{5}{12} + 6 \dfrac{4}{11}\)

Answer

\(\dfrac{1}{2} + 6 \dfrac{1}{2} = 7 (6 \dfrac{103}{132})\)

Exercise \(\PageIndex{10}\)

\(\dfrac{3}{7} + 8 \dfrac{4}{15}\)

Exercise \(\PageIndex{11}\)

\(\dfrac{9}{10} + 2 \dfrac{3}{8}\)

Answer

\(1 + 2 \dfrac{1}{2} = 3 \dfrac{1}{2} (3 \dfrac{11}{40})\)

Exercise \(\PageIndex{12}\)

\(\dfrac{19}{20} + 15 \dfrac{5}{9}\)

Exercise \(\PageIndex{13}\)

\(8 \dfrac{3}{5} + 4 \dfrac{1}{20}\)

Answer

\(8 \dfrac{1}{2} + 4 = 12 \dfrac{1}{2} (12 \dfrac{13}{20})\)

Exercise \(\PageIndex{14}\)

\(5 \dfrac{3}{20} + 2 \dfrac{8}{15}\)

Exercise \(\PageIndex{15}\)

\(9 \dfrac{1}{15} + 6 \dfrac{4}{5}\)

Answer

\(9 + 7 = 16 (15 \dfrac{13}{15})\)

Exercise \(\PageIndex{16}\)

\(7 \dfrac{5}{12} + 10 \dfrac{1}{16}\)

Exercise \(\PageIndex{17}\)

\(3 \dfrac{11}{20} + 2 \dfrac{13}{25} + 1 \dfrac{7}{8}\)

Answer

\(3 \dfrac{1}{2} + 2 \dfrac{1}{2} + 2 = 8\) (7 \(\dfrac{189}{200}\))

Exercise \(\PageIndex{18}\)

\(6 \dfrac{1}{12} + 1 \dfrac{1}{10} + 5 \dfrac{5}{6}\)

Exercise \(\PageIndex{19}\)

\(\dfrac{15}{16} - \dfrac{7}{8}\)

Answer

\(1 - 1 = 0 (\dfrac{1}{16})\)

Exercise \(\PageIndex{20}\)

\(\dfrac{12}{25} - \dfrac{9}{20}\)

Exercises for Review

Exercise \(\PageIndex{21}\)

The fact that

\((\text{a first number } \cdot \text{a second number}) \cdot \text{a third number} = \text{a first number } \cdot (\text{a second number } \cdot \text{a third number})\)

is an example of which property of multiplication?

Answer

associative

Exercise \(\PageIndex{22}\)

Find the quotient: \(\dfrac{14}{15} \div \dfrac{4}{45}\).

Exercise \(\PageIndex{23}\)

Find the difference: \(3 \dfrac{5}{9} - 2 \dfrac{2}{3}\)

Answer

\(\dfrac{8}{9}\)

Exercise \(\PageIndex{24}\)

Find the quotient: \(4.6 \div 0.11\).

Exercise \(\PageIndex{25}\)

Use the distributive property to compute the product: \(25 \cdot 37\).

Answer

\(25(40 - 3) = 1000 - 75 = 925\)