Chapter 1 Simplification Vulgar Fraction Examples
Question 1. From the following table identify the proper fraction, improper fraction, and mixed fraction
WBBSE Class 6 Vulgar Fraction Examples
Identifying the proper fraction, improper fraction, and mixed fraction:
Proper Fraction: \(\frac{1}{5}, \frac{2}{7}, \frac{3}{8}, \frac{6}{13}, \frac{1}{9}, \frac{2}{5}, \frac{5}{9}, \frac{4}{17}, \frac{11}{12}, \frac{3}{7}\)
Improper Fraction: \(\frac{15}{13}, \frac{29}{19}, \frac{23}{17}\)
Mixed Fraction: \(9 \frac{14}{15}, 1 \frac{22}{25}, 11 \frac{1}{9}, 2 \frac{3}{4}, 3 \frac{5}{11}\)
Question 2. Reduce \(6 \frac{7}{19}\) in improper fraction.
Short Questions on Vulgar Fractions
Given:
⇒ \(6 \frac{7}{19}\)
⇒ \(6 \frac{7}{19}=\frac{6 \times 19+7}{19}=\frac{114+7}{19}=\frac{121}{19}\)
Question 3. Express in mixed fraction:
Wbbse Class 6 Maths Solutions :
[Division of 123 by 7]
∴ \(\frac{123}{7}\) = \(17 \frac{4}{7}\)
∴ the required mixed fraction = \(17 \frac{4}{7}\)
Read And Learn More: WBBSE Solutions For Class 6 Maths Chapter 1 Simplification Solved Problems
Question 4. Express with a lowest common denominator:
\(\frac{5}{8}\), \(\frac{7}{12}\).
Common Problems with Vulgar Fractions
Given:
⇒ \(\frac{5}{8}\), \(\frac{7}{12}\).
The denominators of the given fractions are 8 and 12.
Now 24 ÷ 8 = 3 and 24 ÷ 12 = 2
The common denominator for the given fractions will be 24.
For this, it is required to multiply both the numerator and denominator of the first fraction by 3 and to multiply both the numerator and denominator of the second fraction by 2.
∴ \(\frac{5}{8}=\frac{5 \times 3}{8 \times 3}=\frac{15}{24}\) and \(\frac{7}{12}=\frac{7 \times 2}{12 \times 2}=\frac{14}{24}\)
∴ The given two fractions with the lowest common denominator are \(\frac{15}{24}\) and \(\frac{14}{24}\)
Question 5. Express the following fractions with the lowest common numerator:
\(\frac{12}{13}\), \(\frac{18}{23}\)
Wbbse Class 6 Maths Solutions :
Given:
⇒ \(\frac{12}{13}\), \(\frac{18}{23}\)
The numerators of the given fractions are 12 and 18.
∴ The L. C. M. of 12, 18 = 2 x 3 x 2 x 3.36.
The lowest common numerator of the given two fractions will be 36.
Now 36 12 3 and 36 18 = 2
Multiplying both the numerator and denominator of the first fraction by 3 and multiplying both the numerator and denominator of the second fraction by 2, we get,
∴ \(\frac{12}{13}=\frac{12 \times 3}{13 \times 3}=\frac{36}{39}\) and \(\frac{7}{12}=\frac{7 \times 2}{12 \times 2}=\frac{14}{24}\)
∴ The required two fractions with the lowest common numerator are \(\frac{36}{39}\) and \(\frac{36}{46}\)
Question 6. Write 3 equivalent fractions of each of the following fractions:
1. \(\frac{1}{5}\)
Practice Problems on Vulgar Fractions
Given:
⇒ \(\frac{1}{5}\)
We know that, if we multiply both the numerator and denominator of any fraction, then the value of the fraction is not changed.
∴ \(\frac{1}{5}=\frac{1 \times 2}{5 \times 2}=\frac{1 \times 3}{5 \times 3}=\frac{1 \times 4}{5 \times 4}=\cdots \cdots \text { etc. }\)
or, \(\frac{1}{5}=\frac{2}{10}=\frac{3}{15}=\frac{4}{20}=\cdots \cdots \text { etc. }\)
∴ The required three equivalent fractions are \(\frac{2}{10}, \frac{3}{15}, \frac{4}{20}\)
2. \(1 \frac{1}{3}\)
Solution:
Given:
⇒ \(1 \frac{1}{3}\)
⇒ \(1 \frac{1}{3}=\frac{4}{3}=\frac{4 \times 2}{3 \times 2}=\frac{4 \times 3}{3 \times 3}=\frac{4 \times 4}{3 \times 4}=\cdots \cdots \text { etc. }\)
or, \(1 \frac{1}{3}=\frac{8}{6}=\frac{12}{9}=\frac{16}{12}=\cdots \cdots \text { etc. }\)
∴ The required 3 equivalent fractions are [late]\frac{8}{6}, \frac{12}{9}, \frac{16}{12}[/latex]
Question 7. Reduce the following fractions into the lowest terms:
1. \({72}{108}\)
Class 6 Wb Board Math Solution :
⇒ \(\frac{72}{108}\)
2. \(\frac{243}{405}\)
Solution:
⇒ \(\frac{243}{405}\)
Example 8. Arrange the following fractions in ascending order of magnitude:
1. \(\frac{7}{2}, \frac{7}{4}, \frac{7}{5}\)
Solution:
⇒ \(\frac{7}{2}, \frac{7}{4}, \frac{7}{5}\)
The denominators of the fractions are 2, 4, and 5.
∴ L. C. M. of 2, 4, 5 = 2 x 1 x 2 x 5 = 20
20 ÷ 2 = 10 ; 20 ÷ 4 = 5; 20 ÷ 5 = 4.
Multiplying both numerator and denominator of the 1st fraction by 10; the 2nd fraction by 5; the 3rd fraction by 4, we get,
⇒ \(\frac{7}{2}=\frac{7 \times 10}{2 \times 10}=\frac{70}{20} ; \quad \frac{7}{4}=\frac{7 \times 5}{4 \times 5}=\frac{35}{20} ; \quad \frac{7}{5}=\frac{7 \times 4}{5 \times 4}=\frac{28}{20}\)
Now the denominators of all the given fractions are 20.
∵ 28 < 35 < 70.
∴ \(\frac{28}{20}<\frac{35}{20}<\frac{70}{20} \text { or, } \frac{7}{5}<\frac{7}{4}<\frac{7}{2}\)
∴ Arranging the fraction in ascending order, we get \(\frac{7}{5}, \frac{7}{4}, \frac{7}{2}\)
2. \(5 \frac{3}{4}, 5 \frac{5}{9}, 5 \frac{8}{12}\)
Solution:
⇒ \(5 \frac{3}{4}, 5 \frac{5}{9}, 5 \frac{8}{12}\)
Now,
⇒ \(5 \frac{3}{4}=\frac{23}{4} ; \quad 5 \frac{5}{9}=\frac{50}{9} ; \quad 5 \frac{8}{12}=\frac{68}{12}\)
The denominators of the given mixed fractions are 4,9,12.
∴ L. C. M. of 4, 9, 12 = 2 x 2 x 3 x 3 = 36.
36 ÷ 4 = 9; 36 ÷ 9 = 4; 36 ÷ 12 = 3.
then
As 200 < 204 < 207, we get, \(\frac{200}{36}<\frac{204}{36}<\frac{207}{36} \text { or, } 5 \frac{5}{9}<5 \frac{8}{12}<5 \frac{3}{4}\)
∴ Arranging the fractions in ascending order, we get, \(5 \frac{5}{9}, 5 \frac{8}{12}, 5 \frac{3}{4}\)
3. \(1 \frac{1}{5}, 1 \frac{1}{7}, 1 \frac{1}{8}\)
Examples of Real-Life Applications of Vulgar Fractions
⇒ \(1 \frac{1}{5}, 1 \frac{1}{7}, 1 \frac{1}{8}\) Since each of the fractions has the integral part 1, we take the fractions \(\)
The denominator is 5, 7, and 8.
These numbers are prime to each other.
Their L. C. M. = 5 x 7 x 8 = 280.
280 ÷ 5 = 56
280 ÷ 7 = 40
280 ÷ 8 = 35.
∴ \(\frac{1}{5}=\frac{1 \times 56}{5 \times 56}=\frac{56}{280} ; \frac{1}{7}=\frac{1 \times 40}{7 \times 40}=\frac{40}{280} ; \frac{1}{8}=\frac{1 \times 35}{8 \times 35}=\frac{35}{280}\)
As 35 < 40 < 56, we, get \(\frac{35}{280}<\frac{40}{280}<\frac{56}{280} \text { or, } \frac{1}{8}<\frac{1}{7}<\frac{1}{5}\)
∴ \(1 \frac{1}{8}<1 \frac{1}{7}<1 \frac{1}{5}\)
∴ Arranging the fractions in ascending order, we get, \(1 \frac{1}{8}<1 \frac{1}{7}<1 \frac{1}{5}\)
4. \(\frac{1}{3}, \frac{4}{5}, \frac{7}{15}\)
Class 6 Wb Board Math Solution :
⇒ \(\frac{1}{3}, \frac{4}{5}, \frac{7}{15}\) The denominators of the given fractions are 3, 5, 15
∴ L.C.M. of 3, 5, 15= 3 x 5
=15
15 ÷ 3 = 5
15 ÷ 5 = 3
15 ÷ 15 = 1.
∵ 5 < 7 < 12,
∴ \(\frac{5}{15}<\frac{7}{15}<\frac{12}{15} \text { or, } \frac{1}{3}<\frac{7}{15}<\frac{4}{5}\)
∴ Arranging the fractions in ascending order, we get, \(\frac{1}{3}, \frac{7}{15}, \frac{4}{5}\)
5. \(\frac{5}{7}, \frac{3}{4}, \frac{1}{4}\)
Solution:
⇒ \(\frac{5}{7}, \frac{3}{4}, \frac{1}{4}\)
∴ The denominators of the given fractions are 7, 4, and 4.
∴ L. C. M. of 7, 4, 4 = 4 x 7
= 28.
28 ÷ 2 = 45
90 ÷ 9 = 10
90 ÷ 5 = 18.
∵ 7 < 20 < 21,
∴ \(\frac{7}{28}<\frac{20}{28}<\frac{21}{28} \text { or, } \frac{1}{4}<\frac{5}{7}<\frac{3}{4}\)
∴ Arranging the fractions in ascending order, we get, \([latex]\frac{1}{4}, \frac{5}{7}, \frac{3}{4}\)[/latex]
6. \(3 \frac{1}{2}, 7 \frac{5}{9}, 7 \frac{1}{5}\)
Solution:
⇒ \(3 \frac{1}{2}, 7 \frac{5}{9}, 7 \frac{1}{5}\)
Now,
⇒ \(3 \frac{1}{2}\) = \(\frac{7}{2}\)
⇒ \(7 \frac{5}{9}\) = \(\frac{68}{9}\)
⇒ \(7 \frac{1}{5}\) = \(\frac{36}{5}\)
The denominators of the given fractions are 2, 9, and 5.
∴ L. C. M. of 2, 9, 5 = 2 x 9 x 5
= 90. (Here 2, 9, and 5 are prime to each other)
90 ÷ 2 = 45
90 ÷ 9 = 10
90 ÷ 5 = 18.
As 315 648 < 680, we get, \(\frac{315}{90}<\frac{648}{90}<\frac{680}{90} \text { or, } 3 \frac{1}{2}<7 \frac{1}{5}<7 \frac{5}{9}\)
∴ Arranging the fractions in ascending order, we get, \(3 \frac{1}{2}, 7 \cdot \frac{1}{5}, 7 \frac{5}{9}\)
7. \(\frac{1}{8}, \frac{7}{10}, \frac{3}{5}\)
Solution:
⇒ \(\frac{1}{8}, \frac{7}{10}, \frac{3}{5}\) The denominators of the given fractions are 8, 10, and 5.
∴ L. C. M. of 8, 10, 5 = 2 x 5 x 4 = 40.
40 ÷ 8 = 5
40 ÷ 10 = 4
40 ÷ 5 = 8.
∵ 5 < 24, < 28,
∴ \(\frac{5}{40}<\frac{24}{40}<\frac{28}{40} \quad \text { or, } \frac{1}{8}<\frac{3}{5}<\frac{7}{10}\)
∴ Arranging the fractions in ascending order, we get, \(\frac{1}{8}, \frac{3}{5}, \frac{7}{10}\)
3. \(3 \frac{1}{2}, 3 \frac{5}{9}, 3 \frac{1}{5}\)
Solution:
⇒ \(3 \frac{1}{2}, 3 \frac{5}{9}, 3 \frac{1}{5}\)
Since such of the given fractions has the integer 3.
So we have to test the fractions \(\frac{1}{2}, \frac{5}{9}, \frac{1}{5}\)
Now the denominators of the fractions \(\frac{1}{2}, \frac{5}{9}, \frac{1}{5}\) are 2, 9, 5.
L. C M of 2, 9, 5 = 2 x 9 x 5
= 90
∴ 90 ÷ 2 = 45
90 ÷ 9 = 10
90 ÷ 5 = 18
As, 18< 45 < 50;
∴ \(\frac{18}{90}<\frac{45}{90}<\frac{50}{90} \quad \text { or, } \quad \frac{1}{5}<\frac{1}{2}<\frac{5}{9}\)
or, \(3+\frac{1}{5}<3+\frac{1}{2}<3+\frac{5}{9} \quad \text { or, } \quad 3 \frac{1}{5}<3 \frac{1}{2}<3 \frac{5}{9}\)
∴ Arranging the fractions in ascending order, we get, \(3 \frac{1}{5}, 3 \frac{1}{2}, 3 \frac{5}{9}\)
Example 9. Add: \(\frac{7}{2}+\frac{2}{3}+1 \frac{1}{2}\)
Solution:
Given:
⇒ \(\frac{7}{2}+\frac{2}{3}+1 \frac{1}{2}\)
Example 10. Subtract: \(10 \frac{2}{3}-7 \frac{2}{5}\)
Solution:
Given:
⇒ \(10 \frac{2}{3}And7 \frac{2}{5}\)
Example 11. Simplify: \(4 \frac{2}{15}+8 \frac{3}{5}-9 \frac{4}{25}\)
Solution:
The given quality = \(4 \frac{2}{15}+8 \frac{3}{5}-9 \frac{4}{25}\)
The required answer = \(3 \frac{43}{75}\)
Conceptual Questions on Converting Decimals to Vulgar Fractions
Example 12. Find the value of the following:
1. \(\frac{2}{7}-\frac{2}{3}+1 \frac{1}{2}\)
Solution:
⇒ \(\frac{2}{7}-\frac{2}{3}+1 \frac{1}{2}\)
2. \(1 \frac{2}{5}-\frac{3}{8}+\frac{1}{4}\)
Solution:
⇒ \(1 \frac{2}{5}-\frac{3}{8}+\frac{1}{4}\)
[ L.C.M of 5, 8, 4 :
∴ L.C.M. of 5, 8, 4 = 2 x 2 x 5 x 2 = 40]
3. \(\frac{2}{5}+\frac{3}{8}-\frac{1}{4}\)
Solution:
⇒ \(\frac{2}{5}+\frac{3}{8}-\frac{1}{4}\)
4. \(7-3 \frac{1}{8}-2 \frac{1}{3}\)
Solution:
⇒ \(7-3 \frac{1}{8}-2 \frac{1}{3}\)
Real-Life Scenarios Involving Cooking Measurements
5. \(\frac{4}{5}+\frac{5}{8}-1 \frac{1}{3}\)
Solution:
⇒ \(\frac{4}{5}+\frac{5}{8}-1 \frac{1}{3}\)
6. \(1 \frac{3}{10}+1 \frac{4}{5}-1 \frac{1}{4}\)
Solution:
⇒ \(1 \frac{3}{10}+1 \frac{4}{5}-1 \frac{1}{4}\)
∴ L.C.M of 10, 5, 4 = 2 x 5 x 2
= 20.
7. \(2 \frac{5}{6}-1 \frac{8}{9}+1 \frac{3}{4}\)
Solution:
⇒ \(2 \frac{5}{6}-1 \frac{8}{9}+1 \frac{3}{4}\)
∴ L.C.M of 6, 9, 4 = 2 x 3 x 3 x 2
= 36.
8. \(4 \frac{1}{7}+2 \frac{2}{5}-5\)
Solution:
Given
⇒ \(4 \frac{1}{7}+2 \frac{2}{5}-5\)
Example 13. Simplify: \(8 \frac{1}{4} \div 1 \frac{4}{17} \div 5 \frac{7}{8} \text { of } 1 \frac{2}{15}\)
Solution:
Given expression = \(8 \frac{1}{4} \div 1 \frac{4}{17} \div 5 \frac{7}{8} \text { of } 1 \frac{2}{15}\)
∴ The required answer = \(1 \frac{1}{329}\)
Example 14. Simplify: \(1 \frac{1}{4}+\frac{1}{3}\left[2 \frac{1}{4}+1 \frac{1}{2}\left\{3 \frac{1}{2} \div 2 \frac{1}{3}\left(4 \frac{1}{4} \div \overline{2+3 \frac{2}{3}}\right)\right\}\right]\)
Solution:
The given expression = \(1 \frac{1}{4}+\frac{1}{3}\left[2 \frac{1}{4}+1 \frac{1}{2}\left\{3 \frac{1}{2} \div 2 \frac{1}{3}\left(4 \frac{1}{4} \div \overline{2+3 \frac{2}{3}}\right)\right\}\right]\)
∴ The required answer = 3
Example 15. Simplify: \(\frac{4 \frac{1}{2}}{32} \times \frac{2 \frac{2}{3} \div \frac{5}{8}}{1 \frac{1}{5} \text { of } \frac{5}{6} \div 8 \frac{1}{3}} \times \frac{2}{5}\)
Solution:
⇒ \(\frac{4 \frac{1}{2}}{32} \times \frac{2 \frac{2}{3} \div \frac{5}{8}}{1 \frac{1}{5} \text { of } \frac{5}{6} \div 8 \frac{1}{3}} \times \frac{2}{5}\)
∴ The required answer = 2.
Example 16. Simplify: \(\frac{1 \frac{1}{4}-\frac{5}{12}}{1 \frac{1}{4}+\frac{5}{12}}-4 \div \frac{6 \frac{1}{2}}{2+\frac{1}{1+\frac{1}{1+\frac{1}{2}}}}+4 \frac{5}{7 \frac{2}{3}}\)
Solution:
The given expression = \(\frac{1 \frac{1}{4}-\frac{5}{12}}{1 \frac{1}{4}+\frac{5}{12}}-4 \div \frac{6 \frac{1}{2}}{2+\frac{1}{1+\frac{1}{1+\frac{1}{2}}}}+4 \frac{5}{7 \frac{2}{3}}\)