The World of Numbers | Exercise 3.3

Question 2

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}

Check your answer with HomiSolve it yourself, then let Homi check your steps and spot mistakes.
Solution
Understand the Question
  • To add fractions with different denominators, find the Least Common Multiple (LCM) of the denominators to determine a common denominator.
  • Convert each fraction into an equivalent fraction having this common denominator.
  • Add or subtract the numerators while keeping the common denominator, paying careful attention to negative signs.

(i) Find the sum: 25+310\dfrac{2}{5} + \dfrac{3}{10}

Step 1 · Find Common Denominator and Add

LCM of denominators 55 and 1010 is 1010.Diagram 1

Convert 25\dfrac{2}{5} to an equivalent fraction with denominator 1010

25=2×25×2=410\begin{aligned} \dfrac{2}{5} &= \dfrac{2 \times 2}{5 \times 2} \\[0.6em] &= \dfrac{4}{10} \end{aligned}

Now, add the fractions 410+310=4+310=710\dfrac{4}{10} + \dfrac{3}{10} = \dfrac{4 + 3}{10} = \dfrac{7}{10}

Answer

(i) 710\dfrac{7}{10}

(ii) Find the sum: 712+58\dfrac{7}{12} + \dfrac{5}{8}

Step 1 · Find Common Denominator and Add

LCM of denominators 1212 and 88 is 2424.

Convert each fraction to have a denominator of 2424

712=7×212×2=142458=5×38×3=1524\begin{aligned} \dfrac{7}{12} &= \dfrac{7 \times 2}{12 \times 2} = \dfrac{14}{24} \\[0.6em] \dfrac{5}{8} &= \dfrac{5 \times 3}{8 \times 3} = \dfrac{15}{24} \end{aligned}

Now, add the fractions 1424+1524=14+1524=2924\dfrac{14}{24} + \dfrac{15}{24} = \dfrac{14 + 15}{24} = \dfrac{29}{24}

Answer

(ii) 2924\dfrac{29}{24}

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

Step 1 · Find Common Denominator and Add

LCM of denominators 77 and 1414 is 1414.

Convert −47-\dfrac{4}{7} to an equivalent fraction with denominator 1414

−47=−4×27×2=−814\begin{aligned} -\dfrac{4}{7} &= -\dfrac{4 \times 2}{7 \times 2} \\[0.6em] &= -\dfrac{8}{14} \end{aligned}

Now, add the fractions −814+314=−8+314=−514-\dfrac{8}{14} + \dfrac{3}{14} = \dfrac{-8 + 3}{14} = -\dfrac{5}{14}

Answer

(iii) −514-\dfrac{5}{14}

Common Mistakes
  • Direct Addition of Denominators: Adding both numerators and denominators directly (e.g., 25+310=2+35+10\dfrac{2}{5} + \dfrac{3}{10} = \dfrac{2+3}{5+10}) instead of converting to a common denominator first.
  • Sign Errors in Numerator Addition: Forgetting rules for negative integers when evaluating −8+3-8 + 3, which gives −5-5 rather than −11-11 or 55.

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}

← Back to The World of Numbers