Example

Name the fraction of the shape that is shaded in each of the figures. Solution: We need to ask two questions. First, how many equal parts are there? This will be the denominator. Second, of these equal parts, how many are shaded? This will be the numerator. [latex-display]\begin{array}{cccc}\text{How many equal parts are there?}\hfill & & & \text{There are eight equal parts}\text{.}\hfill \\ \text{How many are shaded?}\hfill & & & \text{Five parts are shaded}\text{.}\hfill \end{array}[/latex-display] Five out of eight parts are shaded. Therefore, the fraction of the circle that is shaded is [latex]\Large{\frac{5}{8}}[/latex]. [latex-display]\begin{array}{cccc}\text{How many equal parts are there?}\hfill & & & \text{There are nine equal parts}\text{.}\hfill \\ \text{How many are shaded?}\hfill & & & \text{Two parts are shaded}\text{.}\hfill \end{array}[/latex-display] Two out of nine parts are shaded. Therefore, the fraction of the square that is shaded is [latex]\Large{\frac{2}{9}}[/latex].

Example

Shade [latex]\frac{3}{4}[/latex] of the circle.

Answer: Solution The denominator is [latex]4[/latex], so we divide the circle into four equal parts ⓐ. The numerator is [latex]3[/latex], so we shade three of the four parts ⓑ. [latex]\Large{\frac{3}{4}}[/latex] of the circle is shaded.

Example

Use fraction circles to make wholes using the following pieces:

1. [latex]4[/latex] fourths
2. [latex]5[/latex] fifths
3. [latex]6[/latex] sixths

Answer: Solution

Example

Use fraction circles to make wholes using the following pieces:

1. [latex]3[/latex] halves
2. [latex]8[/latex] fifths
3. [latex]7[/latex] thirds

Answer: Solution 1. [latex]3[/latex] halves make [latex]1[/latex] whole with [latex]1[/latex] half left over. 2. [latex]8[/latex] fifths make [latex]1[/latex] whole with [latex]2[/latex] fifths left over. 3. [latex]7[/latex] thirds make [latex]2[/latex] wholes with [latex]2[/latex] thirds left over.

Example

Name the improper fraction modeled. Then write the improper fraction as a mixed number.

Answer: Solution: Each circle is divided into three pieces, so each piece is [latex]{\Large\frac{1}{3}}[/latex] of the circle. There are four pieces shaded, so there are four thirds or [latex]{\Large\frac{4}{3}}[/latex]. The figure shows that we also have one whole circle and one third, which is [latex]1{\Large\frac{1}{3}}[/latex]. So, [latex]{\Large\frac{4}{3}}=1{\Large\frac{1}{3}}[/latex].

Example

Draw a figure to model [latex]{\Large\frac{11}{8}}[/latex].

Answer: Solution: The denominator of the improper fraction is [latex]8[/latex]. Draw a circle divided into eight pieces and shade all of them. This takes care of eight eighths, but we have [latex]11[/latex] eighths. We must shade three of the eight parts of another circle. So, [latex]{\Large\frac{11}{8}}=1{\Large\frac{3}{8}}[/latex].

Example

Use a model to rewrite the improper fraction [latex]{\Large\frac{11}{6}}[/latex] as a mixed number.

Answer: Solution: We start with [latex]11[/latex] sixths [latex]\left({\Large\frac{11}{6}}\right)[/latex]. We know that six sixths makes one whole.

[latex]{\Large\frac{6}{6}}=1[/latex]

That leaves us with five more sixths, which is [latex]{\Large\frac{5}{6}}[/latex] (11 sixths minus 6 sixths is 5 sixths).

So, [latex]{\Large\frac{11}{6}}=1{\Large\frac{5}{6}}[/latex]

Example

Use a model to rewrite the mixed number [latex]1{\Large\frac{4}{5}}[/latex] as an improper fraction.

Answer: Solution: The mixed number [latex]1{\Large\frac{4}{5}}[/latex] means one whole plus four fifths. The denominator is [latex]5[/latex], so the whole is [latex]{\Large\frac{5}{5}}[/latex]. Together five fifths and four fifths equals nine fifths. So, [latex]1{\Large\frac{4}{5}}={\Large\frac{9}{5}}[/latex]