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Addition and Subtraction of Fractions

🎓 Class 6 Mathematics CBSE Theory Ch 7 — Fractions ⏱ ~35 min
🌐 Language: [gtranslate]

This MCQ module is based on: Addition and Subtraction of Fractions

This mathematics assessment will be based on: Addition and Subtraction of Fractions
Targeting Class 6 level in Fractions, with Basic difficulty.

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7.8 Addition and Subtraction of Fractions

Meena's father has made some chikki. Meena ate \(\frac{1}{2}\) of it and her younger brother ate \(\frac{1}{4}\) of it. How much of the chikki did the two children eat together?

Picture the whole chikki. Cut the half Meena ate into two quarters. Now both portions are measured in the same fractional unit (quarters): \[\frac{1}{2}+\frac{1}{4}=\frac{2}{4}+\frac{1}{4}=\frac{3}{4}.\]

Step 1: Different units Meena ate ½ ¼ Step 2: Same unit (fourths) → add ¼ + ¼ + ¼ = ¾
Adding \(\frac{1}{2}+\frac{1}{4}\) by first rewriting as \(\frac{2}{4}+\frac{1}{4}\).
Brahmagupta's Method (628 CE)
The great Indian mathematician Brahmagupta gave a general method for adding and subtracting fractions in his work Brāhmasphuṭasiddhānta:

Step 1: Find a common fractional unit (a common multiple of denominators).
Step 2: Convert both fractions to equivalent fractions with that unit.
Step 3: Add (or subtract) the numerators; keep the common denominator.

Adding Fractions with the Same Denominator

When denominators match, simply add the numerators: \[\frac{2}{5}+\frac{1}{5}=\frac{3}{5},\qquad \frac{3}{7}+\frac{2}{7}=\frac{5}{7}.\]

Adding Fractions with Different Denominators

Use Brahmagupta's method. Example: \(\frac{2}{3}+\frac{3}{4}\).
LCM of 3 and 4 is 12. Convert: \(\frac{2}{3}=\frac{8}{12}\), \(\frac{3}{4}=\frac{9}{12}\).
Now add: \(\frac{8}{12}+\frac{9}{12}=\frac{17}{12}=1\tfrac{5}{12}\).

Subtraction Using Brahmagupta's Method

Example: \(\frac{5}{6}-\frac{2}{3}\). LCM = 6. Convert \(\frac{2}{3}=\frac{4}{6}\). Subtract: \(\frac{5}{6}-\frac{4}{6}=\frac{1}{6}\).

Another: \(\frac{7}{8}-\frac{3}{8}=\frac{4}{8}=\frac{1}{2}\) (already same unit).

\(\tfrac{7}{8}-\tfrac{3}{8}=\tfrac{4}{8}\) × × × 4 eighths left = \(\tfrac{4}{8}=\tfrac{1}{2}\)
Removing 3 of the 7 shaded parts leaves 4 parts → \(\tfrac{4}{8}=\tfrac{1}{2}\).

Adding and Subtracting Mixed Fractions

Convert each mixed fraction to an improper fraction first, then apply Brahmagupta's method.
Example: \(2\tfrac{1}{3}+1\tfrac{1}{2}=\frac{7}{3}+\frac{3}{2}=\frac{14}{6}+\frac{9}{6}=\frac{23}{6}=3\tfrac{5}{6}\).

Worked Example — Rahim mixes paint
Rahim mixes \(\frac{2}{3}\) litre of yellow paint with \(\frac{3}{4}\) litre of blue paint. Total green paint?
\(\frac{2}{3}+\frac{3}{4}=\frac{8}{12}+\frac{9}{12}=\frac{17}{12}=1\tfrac{5}{12}\) litres.
In-text (p.183): Can you find four different unit fractions that add up to 1?
Answer: Try \(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{12}=\frac{6+3+2+1}{12}=\frac{12}{12}=1\). Nice!
Activity: Unit-Fraction Decomposition
L4 Analyse
Materials: Paper, pencil. No calculator!
Predict: Can you find three different unit fractions that add to exactly 1?
  1. Start with \(\frac{1}{2}\). You need \(\frac{1}{2}\) more.
  2. Try \(\frac{1}{3}\) next. You need \(\frac{1}{6}\) more.
  3. Check: \(\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=\frac{3+2+1}{6}=1\). ✓
  4. Now try a different triple — is there another solution?
There is only one triple of different unit fractions that sum to 1, namely \(\frac{1}{2}+\frac{1}{3}+\frac{1}{6}\). This is because \(\frac{1}{2}\) is forced (else the sum can't reach 1 in three unit fractions), and the other two are then determined uniquely.

Figure it Out (p.181)

Q1. \(\frac{5}{8}-\frac{3}{8}\)
\(=\frac{2}{8}=\frac{1}{4}\).
Q2. \(\frac{7}{9}-\frac{5}{9}\)
\(=\frac{2}{9}\).
Q3. \(\frac{10}{27}-\frac{1}{27}\)
\(=\frac{9}{27}=\frac{1}{3}\).

Competency-Based Questions

Scenario: Geeta buys \(\frac{2}{5}\) m of lace, and Shamim buys \(\frac{3}{4}\) m of the same lace to decorate a tablecloth whose perimeter is 1 m.
Q1. What is the total length of lace they have?
L3 Apply
\(\frac{2}{5}+\frac{3}{4}=\frac{8}{20}+\frac{15}{20}=\frac{23}{20}=1\tfrac{3}{20}\) m.
Q2. Will the lace be enough to cover the 1 m perimeter? Analyse by how much they have extra (or short).
L4 Analyse
\(1\tfrac{3}{20}\) > 1 m, so there is enough. Extra = \(\frac{3}{20}\) m = 15 cm.
Q3. Shamim says: "\(\frac{2}{5}+\frac{3}{4}=\frac{5}{9}\)." Evaluate her working.
L5 Evaluate
Wrong. Adding numerators and denominators separately is a common error. You must first convert to a common denominator. Correct answer: \(\frac{23}{20}\).
Q4. Geeta wants exactly 1 m of lace using two different unit fractions (like \(\frac{1}{2}+\frac{1}{2}\) but the two pieces must differ). Create a pair of different unit fractions whose sum is 1, or show this is impossible.
L6 Create
Impossible with two different unit fractions. Two different unit fractions always sum to less than \(\frac{1}{1}+\frac{1}{2}=\frac{3}{2}\), but pairs that equal exactly 1 must both be \(\frac{1}{2}\). However, three differ­ent unit fractions can sum to 1: \(\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1\).

Assertion–Reason Questions

A: \(\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\).
R: LCM of 2 and 3 is 6, giving \(\frac{3}{6}+\frac{2}{6}=\frac{5}{6}\).
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
(a) — Both true, R correctly explains A.
A: \(\frac{2}{5}+\frac{3}{7}=\frac{5}{12}\).
R: To add fractions, add numerators and add denominators.
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
Both false. Correct answer is \(\frac{14}{35}+\frac{15}{35}=\frac{29}{35}\). Rule for addition is to use a common denominator.
A: \(\frac{5}{8}-\frac{3}{8}=\frac{1}{4}\).
R: Fractions with the same denominator can be subtracted by subtracting numerators.
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
(a) — \(\frac{5-3}{8}=\frac{2}{8}=\frac{1}{4}\). R explains A.

Frequently Asked Questions

How do you add like fractions?
Add the numerators and keep the denominator unchanged. For example, 2/7 plus 3/7 equals 5/7.
How do you add unlike fractions?
Find the LCM of the denominators, rewrite each fraction with that common denominator, then add the numerators. For 1/3 plus 1/4, use 12: 4/12 plus 3/12 equals 7/12.
How do you subtract mixed fractions?
Convert each mixed fraction to an improper fraction, make the denominators equal, subtract the numerators, and convert the result back to a mixed fraction if needed.
What is the rule for subtracting like fractions?
Subtract the numerators and keep the same denominator. Example: 7/9 minus 2/9 equals 5/9.
Why do we need a common denominator before adding fractions?
Fractions can only be combined when the pieces are the same size. Matching the denominators ensures every piece represents the same fractional unit, so adding numerators is valid.
How do you simplify the sum of two fractions?
After adding, find the HCF of the numerator and denominator of the result and divide both by it to reach the simplest form.
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Mathematics Class 6 — Ganita Prakash
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