The World of Numbers | Exercise 3.3

Question 3

Find the difference:

(i) 56−14\dfrac{5}{6} - \dfrac{1}{4}

(ii) 118−34\dfrac{11}{8} - \dfrac{3}{4}

(iii) −79−(−23)-\dfrac{7}{9} - \left(-\dfrac{2}{3}\right)

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Solution
Understand the Question
  • To subtract fractions with different denominators, first find the Least Common Multiple (LCM) of the denominators to write them with a common denominator.
  • Once the denominators are equal, subtract the numerators and keep the common denominator: ad−bd=a−bd\dfrac{a}{d} - \dfrac{b}{d} = \dfrac{a - b}{d}.
  • When subtracting a negative fraction, simplify the signs using −(−x)=+x-(-x) = +x.

(i) 56−14\dfrac{5}{6} - \dfrac{1}{4}

Step 1 · Find Common Denominator and Subtract

The LCM\text{LCM} of the denominators 66 and 44 is 1212.Diagram 1

Convert each fraction to an equivalent fraction with denominator 1212 56=5×26×2=1012\dfrac{5}{6} = \dfrac{5 \times 2}{6 \times 2} = \dfrac{10}{12} 14=1×34×3=312\dfrac{1}{4} = \dfrac{1 \times 3}{4 \times 3} = \dfrac{3}{12}

Now, subtract the fractions

1012−312=10−312=712\begin{aligned} \dfrac{10}{12} - \dfrac{3}{12} &= \dfrac{10 - 3}{12} \\[0.6em] &= \dfrac{7}{12} \end{aligned}
Answer

(i) 712\dfrac{7}{12}

(ii) 118−34\dfrac{11}{8} - \dfrac{3}{4}

Step 1 · Find Common Denominator and Subtract

The LCM\text{LCM} of the denominators 88 and 44 is 88.Diagram 2

Convert 34\dfrac{3}{4} to have a denominator of 88 34=3×24×2=68\dfrac{3}{4} = \dfrac{3 \times 2}{4 \times 2} = \dfrac{6}{8}

Now, subtract from the first fraction

118−68=11−68=58\begin{aligned} \dfrac{11}{8} - \dfrac{6}{8} &= \dfrac{11 - 6}{8} \\[0.6em] &= \dfrac{5}{8} \end{aligned}
Answer

(ii) 58\dfrac{5}{8}

(iii) −79−(−23)-\dfrac{7}{9} - \left(-\dfrac{2}{3}\right)

Step 1 · Simplify Signs and Add Fractions

Simplify the double negative −79−(−23)=−79+23-\dfrac{7}{9} - \left(-\dfrac{2}{3}\right) = -\dfrac{7}{9} + \dfrac{2}{3}Diagram 3

The LCM\text{LCM} of 99 and 33 is 99.

Convert 23\dfrac{2}{3} to have a denominator of 99 23=2×33×3=69\dfrac{2}{3} = \dfrac{2 \times 3}{3 \times 3} = \dfrac{6}{9}

Now, combine the fractions

−79+69=−7+69=−19\begin{aligned} -\dfrac{7}{9} + \dfrac{6}{9} &= \dfrac{-7 + 6}{9} \\[0.6em] &= -\dfrac{1}{9} \end{aligned}
Answer

(iii) −19-\dfrac{1}{9}

Common Mistakes
  • Direct Subtraction: Subtracting numerators and denominators directly (e.g. 56−14≠5−16−4\frac{5}{6} - \frac{1}{4} \ne \frac{5-1}{6-4}). Denominators must be made common using the LCM first.
  • Sign Errors with Negatives: Forgetting that subtracting a negative number is equivalent to addition: −(−23)=+23- \left(-\frac{2}{3}\right) = +\frac{2}{3}.

More questions in Exercise 3.3

Q1

Prove that the following rational numbers are equal:

(i) 23\dfrac{2}{3} and 46\dfrac{4}{6}

(ii) 54\dfrac{5}{4} and 108\dfrac{10}{8}

(iii) −35-\dfrac{3}{5} and −610-\dfrac{6}{10}

(iv) 93\dfrac{9}{3} and 33

Q2

Find the sum:

(i) 25+310\dfrac{2}{5} + \dfrac{3}{10}

(ii) 712+58\dfrac{7}{12} + \dfrac{5}{8}

(iii) −47+314-\dfrac{4}{7} + \dfrac{3}{14}

Q3

Find the difference:

(i) 56−14\dfrac{5}{6} - \dfrac{1}{4}

(ii) 118−34\dfrac{11}{8} - \dfrac{3}{4}

(iii) −79−(−23)-\dfrac{7}{9} - \left(-\dfrac{2}{3}\right)

Q4

Find the product:

(i) 23×310\dfrac{2}{3} \times \dfrac{3}{10}

(ii) 711×58\dfrac{7}{11} \times \dfrac{5}{8}

(iii) −47×514-\dfrac{4}{7} \times \dfrac{5}{14}

Q5

Find the quotient:

(i) 23÷310\dfrac{2}{3} \div \dfrac{3}{10}

(ii) 711÷58\dfrac{7}{11} \div \dfrac{5}{8}

(iii) −47÷514-\dfrac{4}{7} \div \dfrac{5}{14}

Q6

Show that:

(12+34)×83=12×83+34×83\left(\dfrac{1}{2} + \dfrac{3}{4}\right) \times \dfrac{8}{3} = \dfrac{1}{2} \times \dfrac{8}{3} + \dfrac{3}{4} \times \dfrac{8}{3}

Q7

Simplify the following using the distributive property:

79(67−34)\dfrac{7}{9}\left(\dfrac{6}{7} - \dfrac{3}{4}\right)

Q8

Find the rational number xx such that:

56(x+35)=56x+12\dfrac{5}{6}\left(x + \dfrac{3}{5}\right) = \dfrac{5}{6}x + \dfrac{1}{2}

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