Working with Fractions

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CLASS VII Mathematics ~6 marks/year Ch 8 of 15
Working with Fractions

Class 7 · Mathematics · NCERT chapter notes · Akanksha Classes

Snapshot
  • A fraction $\dfrac{p}{q}$ describes a part of a whole; the top is the numerator, the bottom the denominator.
  • Fractions come as proper, improper and mixed; improper and mixed forms convert into each other.
  • Multiplying a fraction by a fraction means "a part of a part": multiply numerators and multiply denominators.
  • Dividing by a fraction means multiplying by its reciprocal (flip the divisor).
  • "Of" means multiply: $\dfrac{3}{4}$ of $20$ is $\dfrac{3}{4} \times 20 = 15$.
  • You add and subtract fractions using a common denominator, and you simplify answers to lowest terms.
  • Weightage: ~6 marks/year — computation of products and quotients, "of" word problems, and simplification.
Detailed Notes

1. What a fraction means

A fraction represents one or more equal parts of a whole. In $\dfrac{p}{q}$, the denominator $q$ tells how many equal parts the whole is divided into, and the numerator $p$ tells how many of those parts we take. So $\dfrac{3}{8}$ means the whole is cut into $8$ equal pieces and we take $3$ of them.

Fraction: $\dfrac{\text{numerator}}{\text{denominator}}$, $\dfrac{p}{q}$ with $q \ne 0$, representing $p$ parts out of $q$ equal parts.

Equivalent fractions name the same amount: $\dfrac{1}{2} = \dfrac{2}{4} = \dfrac{3}{6}$. We get them by multiplying or dividing numerator and denominator by the same non-zero number. A fraction is in lowest terms (simplest form) when the numerator and denominator share no common factor other than $1$.

2. Proper, improper and mixed fractions

Type Description Example
Propernumerator $<$ denominator (less than $1$)$\dfrac{3}{5}$
Impropernumerator $\ge$ denominator (one or more)$\dfrac{7}{5}$
Mixeda whole number plus a proper fraction$1\dfrac{2}{5}$
Worked example — converting between forms

Improper to mixed: $\dfrac{17}{5}$. Divide $17 \div 5 = 3$ remainder $2$, so $\dfrac{17}{5} = 3\dfrac{2}{5}$.

Mixed to improper: $2\dfrac{3}{4} = \dfrac{2 \times 4 + 3}{4} = \dfrac{11}{4}$. Multiply the whole by the denominator, add the numerator, keep the denominator.

3. Multiplying a fraction by a whole number

Multiplying by a whole number is repeated addition. $\dfrac{2}{7} \times 3 = \dfrac{2}{7} + \dfrac{2}{7} + \dfrac{2}{7} = \dfrac{6}{7}$. In short, multiply the numerator by the whole number and keep the denominator.

$\dfrac{p}{q} \times n = \dfrac{p \times n}{q}$, then simplify.
Worked example

$\dfrac{3}{8} \times 4 = \dfrac{3 \times 4}{8} = \dfrac{12}{8} = \dfrac{3}{2} = 1\dfrac{1}{2}$. Always reduce to lowest terms at the end.

4. The meaning of "of" and a fraction of a fraction

The word "of" in fraction problems means multiply. "Half of $10$" is $\dfrac{1}{2} \times 10 = 5$. Taking a fraction of a fraction means a part of a part.

Worked example — a part of a part

$\dfrac{1}{2}$ of $\dfrac{1}{3}$: take a strip, shade $\dfrac{1}{3}$ of it, then take half of that shaded part. The result is $\dfrac{1}{2} \times \dfrac{1}{3} = \dfrac{1 \times 1}{2 \times 3} = \dfrac{1}{6}$. Picturing the strip shows why the answer is smaller than either fraction.

This leads directly to the rule for multiplying any two fractions.

5. Multiplying a fraction by a fraction

$\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d}$ — multiply numerators together and denominators together, then simplify.
Worked example — with cancelling

$\dfrac{4}{9} \times \dfrac{3}{8}$. Multiply across: $\dfrac{4 \times 3}{9 \times 8} = \dfrac{12}{72} = \dfrac{1}{6}$. It is faster to cancel first: $\dfrac{4}{8} = \dfrac{1}{2}$ and $\dfrac{3}{9} = \dfrac{1}{3}$, giving $\dfrac{1}{2} \times \dfrac{1}{3} = \dfrac{1}{6}$.

Worked example — mixed numbers

$1\dfrac{1}{2} \times 2\dfrac{2}{3}$. First convert to improper: $\dfrac{3}{2} \times \dfrac{8}{3} = \dfrac{3 \times 8}{2 \times 3} = \dfrac{24}{6} = 4$. Always convert mixed numbers to improper fractions before multiplying.

Key observation: when you multiply a number by a proper fraction (less than $1$), the result is smaller than the number; when you multiply by an improper fraction (greater than $1$), the result is larger.

6. The reciprocal of a fraction

The reciprocal (or multiplicative inverse) of a fraction is what you get by turning it upside down. The reciprocal of $\dfrac{3}{5}$ is $\dfrac{5}{3}$. A fraction times its reciprocal always equals $1$.

Reciprocal of $\dfrac{a}{b}$ is $\dfrac{b}{a}$ (provided $a \ne 0$); and $\dfrac{a}{b} \times \dfrac{b}{a} = 1$.

The reciprocal of a whole number $n$ is $\dfrac{1}{n}$, since $n = \dfrac{n}{1}$. The number $0$ has no reciprocal because we cannot divide by $0$.

7. Dividing fractions

Dividing by a fraction is the same as multiplying by its reciprocal. Asking "how many $\dfrac{1}{4}$s are in $2$?" is $2 \div \dfrac{1}{4} = 2 \times \dfrac{4}{1} = 8$ — and indeed there are eight quarter-pieces in $2$ wholes.

$\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}$ — flip the divisor and multiply.
Worked example — fraction by fraction

$\dfrac{3}{5} \div \dfrac{9}{10} = \dfrac{3}{5} \times \dfrac{10}{9} = \dfrac{3 \times 10}{5 \times 9} = \dfrac{30}{45} = \dfrac{2}{3}$.

Worked example — whole number by fraction

$6 \div \dfrac{3}{4} = 6 \times \dfrac{4}{3} = \dfrac{24}{3} = 8$.

Worked example — fraction by whole number

$\dfrac{4}{5} \div 2 = \dfrac{4}{5} \times \dfrac{1}{2} = \dfrac{4}{10} = \dfrac{2}{5}$.

8. Adding and subtracting fractions

To add or subtract, the fractions must have the same denominator. If they do, just add or subtract the numerators. If not, rewrite them with a common denominator (the LCM of the denominators) first.

Worked example — unlike denominators

$\dfrac{2}{3} + \dfrac{1}{4}$. LCM of $3$ and $4$ is $12$. Rewrite: $\dfrac{2}{3} = \dfrac{8}{12}$ and $\dfrac{1}{4} = \dfrac{3}{12}$. Add: $\dfrac{8}{12} + \dfrac{3}{12} = \dfrac{11}{12}$.

Worked example — mixed-number subtraction

$3\dfrac{1}{2} - 1\dfrac{3}{4} = \dfrac{7}{2} - \dfrac{7}{4} = \dfrac{14}{4} - \dfrac{7}{4} = \dfrac{7}{4} = 1\dfrac{3}{4}$.

9. Word problems with fractions

Worked example — sharing

A ribbon $\dfrac{3}{4}$ m long is cut into pieces each $\dfrac{1}{8}$ m long. How many pieces? Divide: $\dfrac{3}{4} \div \dfrac{1}{8} = \dfrac{3}{4} \times \dfrac{8}{1} = \dfrac{24}{4} = 6$ pieces.

Worked example — "of" a quantity

A jug holds $2\dfrac{1}{2}$ litres of juice. Riya drinks $\dfrac{2}{5}$ of it. How much did she drink? $\dfrac{2}{5} \times \dfrac{5}{2} = \dfrac{10}{10} = 1$ litre.

10. Common mistakes to avoid

  • Adding numerators and denominators directly: $\dfrac{1}{2} + \dfrac{1}{3} \ne \dfrac{2}{5}$. You need a common denominator.
  • Forgetting to convert mixed numbers to improper form before multiplying or dividing.
  • Flipping the wrong fraction in division — always flip the divisor (the second one).
  • Leaving the answer unsimplified.
  • Thinking multiplying always makes a number bigger — a proper fraction makes it smaller.

11. Quick revision checklist

  • Fraction $= \dfrac{\text{numerator}}{\text{denominator}}$; reduce to lowest terms.
  • Convert between improper and mixed forms freely.
  • Multiply: numerator $\times$ numerator over denominator $\times$ denominator; cancel first if possible.
  • "Of" means multiply.
  • Divide: multiply by the reciprocal of the divisor.
  • Add/subtract only with a common denominator.
Practice MCQs
1. In the fraction $\dfrac{5}{9}$, the number $9$ is the:
  1. numerator
  2. denominator
  3. reciprocal
  4. quotient
Answer: (B) The bottom number is the denominator.
2. Which of these is a proper fraction?
  1. $\dfrac{7}{4}$
  2. $\dfrac{3}{5}$
  3. $\dfrac{9}{9}$
  4. $\dfrac{11}{8}$
Answer: (B) Numerator $<$ denominator, so $\dfrac{3}{5}$ is proper.
3. $\dfrac{17}{5}$ written as a mixed number is:
  1. $3\dfrac{1}{5}$
  2. $3\dfrac{2}{5}$
  3. $2\dfrac{3}{5}$
  4. $3\dfrac{3}{5}$
Answer: (B) $17 \div 5 = 3$ remainder $2$, so $3\dfrac{2}{5}$.
4. $2\dfrac{3}{4}$ as an improper fraction is:
  1. $\dfrac{9}{4}$
  2. $\dfrac{11}{4}$
  3. $\dfrac{14}{4}$
  4. $\dfrac{5}{4}$
Answer: (B) $2 \times 4 + 3 = 11$, so $\dfrac{11}{4}$.
5. $\dfrac{2}{3} \times \dfrac{6}{7} =$
  1. $\dfrac{8}{10}$
  2. $\dfrac{4}{7}$
  3. $\dfrac{12}{21}$ only
  4. $\dfrac{12}{10}$
Answer: (B) $\dfrac{12}{21} = \dfrac{4}{7}$ in lowest terms.
6. "$\dfrac{3}{4}$ of $20$" equals:
  1. $12$
  2. $15$
  3. $16$
  4. $60$
Answer: (B) $\dfrac{3}{4} \times 20 = 15$.
7. The reciprocal of $\dfrac{7}{9}$ is:
  1. $\dfrac{7}{9}$
  2. $\dfrac{9}{7}$
  3. $-\dfrac{7}{9}$
  4. $1$
Answer: (B) Turn it upside down: $\dfrac{9}{7}$.
8. $6 \div \dfrac{3}{4} =$
  1. $\dfrac{9}{2}$
  2. $8$
  3. $\dfrac{18}{4}$
  4. $2$
Answer: (B) $6 \times \dfrac{4}{3} = 8$.
9. $\dfrac{4}{5} \div 2 =$
  1. $\dfrac{8}{5}$
  2. $\dfrac{2}{5}$
  3. $\dfrac{4}{10}$ only
  4. $\dfrac{5}{8}$
Answer: (B) $\dfrac{4}{5} \times \dfrac{1}{2} = \dfrac{4}{10} = \dfrac{2}{5}$.
10. $\dfrac{2}{3} + \dfrac{1}{4} =$
  1. $\dfrac{3}{7}$
  2. $\dfrac{11}{12}$
  3. $\dfrac{3}{12}$
  4. $\dfrac{5}{7}$
Answer: (B) Common denominator $12$: $\dfrac{8}{12} + \dfrac{3}{12} = \dfrac{11}{12}$.
11. Multiplying a whole number by a proper fraction gives a result that is:
  1. larger than the whole number
  2. smaller than the whole number
  3. equal to it
  4. always a whole number
Answer: (B) A proper fraction is less than $1$, so the product is smaller.
12. A fraction multiplied by its reciprocal equals:
  1. $0$
  2. $1$
  3. the fraction itself
  4. $2$
Answer: (B) e.g. $\dfrac{3}{5} \times \dfrac{5}{3} = 1$.
13. How many pieces of $\dfrac{1}{8}$ m can be cut from $\dfrac{3}{4}$ m of ribbon?
  1. $4$
  2. $6$
  3. $8$
  4. $3$
Answer: (B) $\dfrac{3}{4} \div \dfrac{1}{8} = \dfrac{3}{4} \times 8 = 6$.
14. $1\dfrac{1}{2} \times 2\dfrac{2}{3} =$
  1. $3$
  2. $4$
  3. $\dfrac{7}{2}$
  4. $5$
Answer: (B) $\dfrac{3}{2} \times \dfrac{8}{3} = \dfrac{24}{6} = 4$.
15. The simplest form of $\dfrac{12}{72}$ is:
  1. $\dfrac{1}{6}$
  2. $\dfrac{1}{5}$
  3. $\dfrac{2}{12}$
  4. $\dfrac{6}{36}$
Answer: (A) Divide top and bottom by $12$: $\dfrac{1}{6}$.
Important Questions
Q1. Convert $\dfrac{23}{6}$ to a mixed number and $3\dfrac{4}{5}$ to an improper fraction. (2 marks)
Answer: $23 \div 6 = 3$ remainder $5$, so $\dfrac{23}{6} = 3\dfrac{5}{6}$. And $3\dfrac{4}{5} = \dfrac{3 \times 5 + 4}{5} = \dfrac{19}{5}$.
Q2. Evaluate $\dfrac{5}{8} \times \dfrac{4}{15}$, giving the answer in lowest terms. (2 marks)
Answer: $\dfrac{5 \times 4}{8 \times 15} = \dfrac{20}{120} = \dfrac{1}{6}$. (Cancelling first: $\dfrac{5}{15} = \dfrac{1}{3}$ and $\dfrac{4}{8} = \dfrac{1}{2}$, giving $\dfrac{1}{6}$.)
Q3. Explain why dividing by a fraction is the same as multiplying by its reciprocal, with an example. (3 marks)
Answer: Dividing by $\dfrac{c}{d}$ asks how many $\dfrac{c}{d}$-sized parts fit in a quantity. Since $\dfrac{c}{d} \times \dfrac{d}{c} = 1$, multiplying by the reciprocal undoes the divisor. For example $2 \div \dfrac{1}{4} = 2 \times \dfrac{4}{1} = 8$, and indeed there are $8$ quarter-pieces in $2$ wholes.
Q4. A tank holds $4\dfrac{1}{2}$ litres of water. $\dfrac{2}{3}$ of it is used. How much water is used? (2 marks)
Answer: $\dfrac{2}{3} \times 4\dfrac{1}{2} = \dfrac{2}{3} \times \dfrac{9}{2} = \dfrac{18}{6} = 3$ litres are used.
Q5. Evaluate $\dfrac{7}{8} - \dfrac{1}{3} + \dfrac{1}{2}$. (3 marks)
Answer: LCM of $8, 3, 2$ is $24$. $\dfrac{7}{8} = \dfrac{21}{24}$, $\dfrac{1}{3} = \dfrac{8}{24}$, $\dfrac{1}{2} = \dfrac{12}{24}$. So $\dfrac{21}{24} - \dfrac{8}{24} + \dfrac{12}{24} = \dfrac{25}{24} = 1\dfrac{1}{24}$.
Q6. A wire $\dfrac{9}{10}$ m long is cut into pieces each $\dfrac{3}{20}$ m long. Find the number of pieces. (2 marks)
Answer: $\dfrac{9}{10} \div \dfrac{3}{20} = \dfrac{9}{10} \times \dfrac{20}{3} = \dfrac{180}{30} = 6$ pieces.
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