The fractions having the same value are called equivalent fractions. Their numerator and denominator can be different but, they represent the same part of a whole.
Consider the following:
(i)
1/2
(ii)
2/4
(iii)
4/8
(iv)
8/16
(v)
5/10
(vi)
10/20
(viii)
6/12
(viii)
3/6
We can see the shade portion with respect to the whole shape in the figures from (i) to (viii)
In; (i) Shaded to whole shape is half to whole i.e., ^{1}/_{2}Thus, 1/2, 2/4, 4/8, 8/16, 5/10, 10/20, 6/12, 3/6, etc., each fraction represents half portion of the shape, which are all equal. All have different numerator and denominator but they all have the same value because they represent the same shaded area i.e., half of the rectangle.
So, 1/2, 2/4, 4/8, 8/16, 5/10, 10/20, 6/12, 3/6 are equivalent fractions.
We can express it as, 1/2 = 2/4 = 4/8 = 8/16 = 5/10 = 10/20 = 6/12 = 3/6 = 1/2.
The
fractions having different numerators and denominators but representing
equal value or magnitude are called equivalent fractions.
Note:
The fraction 1/2 and 2/4 and 4/8 show the same amount of shaded or colored parts. 1/2 and 2/4 and 4/8 are equivalent fractions.
Equivalent fractions are fractions that have different forms but the same value.
Building Equivalent Fractions:
1. Change 2/5 to an equivalent fraction with denominator 15.
Note:
Multiply numerator and denominator by the same number to get the required denominator.
2. Change 9/12 to an equivalent fraction with denominator 4.
Note:
To find an equivalent fraction with smaller denominator, you can divide the numerator and denominator with the same number.
3. We can build equivalent fraction with multiples of numerator and denominator.
Write the next three equivalent fractions.
Note:
Equivalent fractions have the same value.
Equivalent fraction can be built to very large numbers.
Equivalent fraction can be reduced to the lowest term.
Look at the following:
Look at the figures above.
In A the three different fractions \(\frac{1}{3}\), \(\frac{2}{6}\)
and \(\frac{4}{12}\) represent the same shaded part.
In B the three different fractions \(\frac{1}{4}\), \(\frac{2}{8}\) and \(\frac{4}{16}\) represent the same shaded part.
That is,
\(\frac{1}{3}\) = \(\frac{2}{6}\) = \(\frac{4}{12}\);
\(\frac{1}{4}\) = \(\frac{2}{8}\) = \(\frac{4}{16}\)
Fractions which represent the same part of a whole thing or which indicates the same number of things in a group are called ‘Equivalent Fractions’.
When we multiply the numerator and denominator of a fraction by the same number (other than zero) we get an equivalent fraction with a higher numerator and denominator.
For example;
\(\frac{1 × 2}{6 × 2}\) = \(\frac{2}{12}\)
\(\frac{1 × 3}{6 × 3}\) = \(\frac{3}{18}\)
\(\frac{1 × 4}{6 × 4}\) = \(\frac{4}{24}\)
\(\frac{1}{6}\) = \(\frac{2}{12}\) = \(\frac{3}{18}\) = \(\frac{4}{24}\)
When we divide the numerator and denominator of a fraction by the same number (other than zero) we get an equivalent fraction with a lower numerator and denominator.
For example:
1. \(\frac{36 ÷ 2}{48 ÷ 2}\) = \(\frac{18}{24}\)
\(\frac{18 ÷ 3}{24 ÷ 3}\) = \(\frac{6}{8}\)
\(\frac{6 ÷ 2}{8 ÷ 2}\) = \(\frac{3}{4}\)
\(\frac{3}{4}\) = \(\frac{6}{8}\) = \(\frac{18}{24}\) = \(\frac{36}{48}\)
2. \(\frac{4}{6}\) = \(\frac{.......}{18}\)
Find the relation between the denominators.
6 × ....... = 18 (since 6 < 18)
6 × 3 = 18
Therefore, \(\frac{4 × 3}{6 × 3}\) = \(\frac{12}{18}\)
Therefore, \(\frac{4}{6}\) = \(\frac{12}{18}\)
3. \(\frac{9}{10}\) = \(\frac{27}{.......}\)
Find the relation between the numerators.
6 × ....... = 27 ? (since 9 < 27)
9 × 3 = 27
Therefore, \(\frac{9 × 3}{10 × 3}\) = \(\frac{27}{30}\)
Therefore, \(\frac{9}{10}\) = \(\frac{27}{30}\)
4. \(\frac{24}{32}\) = \(\frac{.......}{8}\)
32 ÷ 8 = 4 (since 32 >8)
\(\frac{24 ÷ 4}{32 ÷ 4}\) = \(\frac{6}{8}\)
Therefore, \(\frac{24}{32}\) = \(\frac{6}{8}\)
Checking for Equivalence of Fractions:
Cross Product Rule:
If the cross products of two fractions are equal, then they are equivalent fractions.
For example:
1. Is \(\frac{2}{3}\) equivalent to \(\frac{4}{6}\)?
2 × 6 = 12
4 × 3 = 12
Therefore, \(\frac{2}{3}\) is equivalent to \(\frac{4}{6}\)
2. Is \(\frac{2}{4}\) equivalent to \(\frac{3}{5}\)?
2 × 5 = 10
3 × 4 = 12
Therefore, \(\frac{2}{4}\) is not equivalent to \(\frac{3}{5}\)
Questions and answers on Equivalent Fractions:
I. Find 4 equivalent fractions for the given fractions by multiplying.
(i) \(\frac{3}{7}\)
(ii) \(\frac{2}{9}\)
(iii) \(\frac{4}{5}\)
(vi) \(\frac{7}{11}\)
Answers:
I. (i) \(\frac{6}{14}\), \(\frac{9}{21}\), \(\frac{12}{28}\), \(\frac{15}{35}\)
(ii) \(\frac{4}{18}\), \(\frac{6}{27}\), \(\frac{8}{36}\), \(\frac{10}{45}\)
(iii) \(\frac{8}{10}\), \(\frac{12}{15}\), \(\frac{16}{20}\), \(\frac{20}{25}\)
(vi) \(\frac{14}{22}\), \(\frac{21}{33}\), \(\frac{28}{44}\), \(\frac{35}{55}\)
II. Fill the boxes to make equivalent fractions:
(i) \(\frac{3}{4}\) = \(\frac{……}{16}\)
(ii) \(\frac{5}{9}\) = \(\frac{35}{……}\)
(iii) \(\frac{7}{8}\) = \(\frac{……}{64}\)
(vi) \(\frac{7}{……}\) = \(\frac{63}{99}\)
(v) \(\frac{2}{13}\) = \(\frac{……}{51}\)
(vi) \(\frac{11}{17}\) = \(\frac{……}{51}\)
Answers:
II. (i) 12
(ii) 63
(iii) 56
(vi) 11
(v) 8
(vi) 33
III. Write two equivalent fractions for the following.
(i) \(\frac{2}{3}\), ........., .........
(ii) \(\frac{4}{5}\), ........., .........
(iii) \(\frac{6}{7}\), ........., .........
(iv) \(\frac{1}{5}\), ........., .........
(v) \(\frac{3}{8}\), ........., .........
(vi) \(\frac{5}{10}\), ........., .........
Answer:
III. (i) \(\frac{4}{6}\), \(\frac{6}{9}\)
(ii) \(\frac{8}{10}\), \(\frac{20}{25}\)
(iii) \(\frac{24}{28}\), \(\frac{36}{42}\)
(iv) \(\frac{3}{15}\), \(\frac{10}{50}\)
(v) \(\frac{9}{24}\), \(\frac{21}{56}\)
(vi) \(\frac{5}{10}\), \(\frac{3}{8}\)
IV. Find the missing terms of the following fractions.
(i) \(\frac{1}{7}\) = \(\frac{5}{.......}\)
(ii) \(\frac{3}{4}\) = \(\frac{12}{.......}\)
(iii) \(\frac{25}{100}\) = \(\frac{1}{.......}\)
(iv) \(\frac{55}{100}\) = \(\frac{.......}{20}\)
(v) \(\frac{3}{7}\) = \(\frac{.......}{63}\)
(vi) \(\frac{5}{6}\) = \(\frac{.......}{24}\)
(vii) \(\frac{6}{11}\) = \(\frac{18}{.......}\)
(viii) \(\frac{15}{48}\) = \(\frac{.......}{16}\)
(ix) \(\frac{25}{40}\) = \(\frac{.......}{8}\)
Answer:
IV. (i) 35
(ii) 16
(iii) 4
(iv) 11
(v) 27
(vi) 20
(vii) 33
(viii) 5
(ix) 5
V. Find an equivalent fraction of \(\frac{3}{4}\) with
(i) Numerator 27
(ii) Denominator 28
(iii) Numerator 30
(iv) Denominator 12
Answer:
V. (i) \(\frac{27}{36}\)
(ii) \(\frac{21}{28}\)
(iii) \(\frac{30}{40}\)
(iv) \(\frac{9}{12}\)
VI. Find an equivalent fraction of \(\frac{54}{60}\) with
(i) Numerator 27
(ii) Denominator 10
(iii) Numerator 9
(iv) Denominator 20
Answer:
VI. (i) \(\frac{27}{30}\)
(ii) \(\frac{9}{10}\)
(iii) \(\frac{9}{10}\)
(iv) \(\frac{18}{20}\)
VII. Indicate which of the following pairs of fractions are equivalent:
(i) \(\frac{3}{5}\) and \(\frac{9}{15}\)
(ii) \(\frac{2}{8}\) and \(\frac{10}{40}\)
(iii) \(\frac{5}{7}\) and \(\frac{25}{42}\)
(iv) \(\frac{9}{11}\) and \(\frac{27}{34}\)
(v) \(\frac{4}{13}\) and \(\frac{12}{39}\)
Answer:
VII: (i) Equivalent Fraction
(ii) Equivalent Fraction
(iii) Not Equivalent Fraction
(iv) Not Equivalent Fraction
(v) Equivalent Fraction
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