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Equivalent Fractions: Problems with Solutions
By Catalin David
Problem 1
Are the fractions pictured in the two drawings equivalent?
Solution:
The fraction pictured in the first drawing is [tex] \frac{2}{4}[/tex] and the fraction pictured in the second drawing is [tex] \frac{1}{2} [/tex]. The two fractions are equivalent because the blue zones are equal.
The numerator and the denominator of the first fraction are two times greater than the numerator and the denominator of the second fraction. At the same time, 2 x 2 = 1 x 4
Problem 2
John cut his pizza into 6 equal slices and ate two of them. Tim cut his pizza(the same size) into 3 equal slices and ate one of them. Did they eat the same amount of pizza?
Solution:
These drawings show the slices eaten by the two of them. John ate [tex] \frac{2}{6}[/tex] of the pizza, while Tim ate [tex]\frac{1}{3}[/tex] of the pizza. The two fractions are equivalent so both of them eat the same amount.
Problem 3
Are the fractions pictured in the two drawings equivalent?
Solution:
The fraction pictured in the first drawing is [tex]\frac{2}{6}[/tex] and the fraction pictured in the second drawing is [tex]\frac{1}{3}[/tex]. The two fractions are equivalent because the numerator and the denominator of the first fraction are two times greater than the numerator and the denominator of the second fraction.
Problem 4
Are the fractions pictured in the two drawings equivalent?
Solution:
The fraction pictured in the first drawing is [tex]\frac{3}{4}[/tex] and the fraction pictured in the second drawing is [tex]\frac{1}{2}[/tex]. The two fractions are not equivalent because the red zones are not equal and 3 × 2 = 6 and 4 × 1 = 4
Problem 5
Are the fractions pictured in the two drawings equivalent?
Solution:
The fraction pictured in the first drawing is [tex] \frac {3}{6}[/tex] and the fraction pictured in the second drawing is[tex] \frac {4}{8} [/tex]. The two fractions are equivalent because 3 × 8 = 6 × 4 = 24
Problem 6
Are the fractions [tex] \frac{2}{3}[/tex] and [tex]\frac {6}{7} [/tex] equivalent?
Solution:
The numerator of the second fraction is 3 times greater than the numerator of the first fraction. The denominator of the second fraction is not 3 times greater than the denominator of the first fraction. In this case, they're not equivalent.
Problem 7
Are the fractions [tex]\frac{9}{6}[/tex] and [tex]\frac{6}{4}[/tex] equivalent?
Solution:
We verify if, in this case, the condition of equivalence of two fractions is true. If [tex]\frac{a}{b} = \frac{c}{d}[/tex] then a × d = b × c . In our case, 9 × 4 = 6 × 6 = 36 Then the fractions are equivalent.
Problem 8
The fraction [tex] \frac {3}{5}[/tex] is equivalent to
[tex]\frac{6}{8}[/tex]
[tex]\frac{6}{10}[/tex]
[tex]\frac{3}{10}[/tex]
[tex]\frac{9}{10}[/tex]
Solution:
[tex]\frac{6}{10}[/tex] because the numerator and the denominator of the second fraction are two times greater than the numerator and the denominator of [tex]\frac {3}{5}[/tex].
Problem 9
Which fraction is equivalent to [tex]\frac{21}{12}[/tex]?
[tex]\frac{7}{12}[/tex]
[tex]\frac{21}{4}[/tex]
[tex]\frac{7}{4}[/tex]
[tex]\frac{7}{2}[/tex]
Solution:
[tex]\frac{7}{4}[/tex] because the numerator and the denominator of the second fraction are three times smaller than the numerator and the denominator of the first fraction.
Problem 10
Which two fractions are equivalent to [tex]\frac{4}{6}[/tex]
[tex]\frac{3}{2}, \frac{12}{18}[/tex]
[tex]\frac{8}{10}, \frac{12}{18}[/tex]
[tex]\frac{2}{4}, \frac{1}{4}[/tex]
[tex]\frac{2}{3}, \frac{12}{18}[/tex]
Solution:
To find fractions equivalent to the given one, it's enough to multiply or divide the numerator and the denominator by the same number. 4÷2 = 2, 6÷2=3, so a fraction equivalent with [tex] \frac {4}{6}[/tex] is [tex] \frac {2}{3}[/tex]. 4 × 3 = 12, 6 × 3 = 18, so another fraction equivalent with [tex] \frac {4}{6}[/tex] is [tex] \frac {12}{18}[/tex].
Problem 11
Can 3, 5, 6, and 10 form 2 equivalent fractions?
Solution:
4 numbers form 2 equivalent fractions if the results of the multiplication of each numerator with the denominator of the other fraction are equal. We observe that 3 x 10 = 5 x 6. The fractions are [tex] \frac {3}{5} = \frac {6}{10} [/tex]
Problem 12
Create 2 equivalent fractions with the numbers 4, 9, 2, and 18
Solution:
We're searching for pairs of numbers which, when multiplied, have the same product. We observe that 4 x 9 = 18 x 2 = 36, so the fractions will be [tex] \frac {2}{9} and \frac {4}{18} [/tex]
Problem 13
Calculate the value of x,
if $\frac{2}{5} = \frac{4}{x}$
Solution:
$\frac{2}{5} = \frac{4}{10}$
$\frac{4}{10} = \frac{4}{x}$ => $x = 10$
Problem 14
If [tex]\frac{1}{4}=\frac {a}{12}[/tex], then
a
=
Solution:
If [tex]\frac{a}{b} = \frac{c}{d} [/tex] then a × d = b × c. In our case, 1 × 12 = 4 × a
12 = 4 × a
a = 12 ÷ 4 = 3
Another solution:
We notice that the denominator of the second fraction is three times greater than the denominator of the first fraction. Thus, the numerator of the second fraction is three times greater than the numerator of the first fraction. Consequently, a = 3.
Problem 15
If [tex]\frac{5}{a} = \frac{20}{28}[/tex] then
a
is
Solution:
5 is 4 times less than 20, so
a
is 4 times less than 28
a = 28 ÷ 4 = 7
Problem 16
If [tex] \frac {a}{3} = \frac {4}{6} [/tex], then
a
=
Solution:
a × 6 = 3 × 4
a × 6 = 12
a = 12 ÷ 6 = 2
Problem 17
If [tex] \frac {6}{2} = \frac {15}{a} [/tex], then
a
is
Solution:
6 × a = 2 × 15
6 × a = 30
a = 30 ÷ 6 = 5
Problem 18
If the fraction [tex]\frac{9}{6}[/tex] is equivalent to [tex]\frac{6}{a}[/tex], then a is
2
4
3
1
Solution:
If the fractions are equivalent, then 9 × a = 36
a = 36 ÷ 9 = 4
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