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Equivalent Fractions and Comparison

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

This MCQ module is based on: Equivalent Fractions and Comparison

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

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7.6 Equivalent Fractions

When the same amount is shown using different fractional units, we get different-looking fractions that are actually equal. Such fractions are called equivalent fractions?.

For example, \(\frac{1}{2}\) of a roti is the same amount as \(\frac{2}{4}\) of the roti, which is the same as \(\frac{3}{6}\) and \(\frac{4}{8}\). So: \[\frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\frac{4}{8}=\frac{5}{10}=\cdots\]

½ ²⁄₄ ³⁄₆ ⁴⁄₈
Fraction Wall: \(\frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\frac{4}{8}\) — the shaded length is the same.
Rule for Equivalent Fractions
Multiplying (or dividing) both the numerator and denominator of a fraction by the same non-zero number gives an equivalent fraction: \[\frac{a}{b}=\frac{a\times k}{b\times k}=\frac{a\div m}{b\div m}\] The value stays the same because we are cutting each piece into more (or fewer) sub-pieces, and taking a matching count.

Simplest Form (Lowest Terms)

A fraction is in lowest terms? when the numerator and denominator share no common factor other than 1. Example: \(\frac{4}{8}=\frac{1}{2}\) after dividing both by 4. Here \(\frac{1}{2}\) is in lowest terms.

Figure it Out (p.170)

Q1. Are \(\frac{3}{6}, \frac{4}{8}, \frac{5}{10}\) equivalent? Why?
Yes — each simplifies to \(\frac{1}{2}\).
Q2. Write two equivalent fractions for \(\frac{2}{6}\).
\(\frac{2}{6}=\frac{4}{12}=\frac{6}{18}=\frac{1}{3}\).
Q3. Fill in: \(\frac{3}{5} = \frac{\square}{10} = \frac{9}{\square} = \frac{\square}{25}\).
\(\frac{6}{10}=\frac{9}{15}=\frac{15}{25}\).

7.7 Comparing Fractions

Comparing two fractions is easy when they share the same denominator (same fractional unit).

Rule 1 — Same Denominator
If two fractions have the same denominator, the one with the larger numerator is larger.
Example: \(\frac{5}{7} > \frac{4}{7}\).
Rule 2 — Same Numerator
If two fractions have the same numerator, the one with the smaller denominator is larger (bigger pieces).
Example: \(\frac{3}{4} > \frac{3}{10}\) because quarters are bigger than tenths.

Comparing Using Equivalent Fractions

To compare fractions with different numerators and denominators, rewrite them with the same denominator (a common multiple of the two denominators), then compare numerators.

Example: Compare \(\frac{3}{4}\) and \(\frac{10}{20}\) (Shabnam's chikki vs Mukta's). LCM of 4 and 20 is 20.
\(\frac{3}{4}=\frac{15}{20}\). Now \(\frac{15}{20} > \frac{10}{20}\), so \(\frac{3}{4} > \frac{10}{20}\).

Fraction 1Fraction 2Common Denom.Result
\(\frac{2}{3}\)\(\frac{3}{5}\)15\(\frac{10}{15} > \frac{9}{15}\)
\(\frac{4}{7}\)\(\frac{5}{7}\)7\(\frac{4}{7} < \frac{5}{7}\)
\(\frac{3}{4}\)\(\frac{5}{8}\)8\(\frac{6}{8} > \frac{5}{8}\)
In-text Q: Which is easier to compare — \(\frac{3}{4}\) and \(\frac{10}{20}\), or \(\frac{4}{7}\) and \(\frac{5}{7}\)?
Answer: The second pair, because they already share the same denominator (7).
Activity: Fraction Wall Builder
L3 Apply
Materials: 10 identical paper strips, scissors, ruler.
Predict: How many \(\frac{1}{6}\) pieces equal three \(\frac{1}{2}\) pieces?
  1. Take one strip = 1 whole.
  2. Fold another strip into halves, another into thirds, another into fourths, sixths, tenths.
  3. Cut them out. Line up the pieces under the whole.
  4. Compare: which stack reaches the same point as 3 × \(\frac{1}{2}\)?
Three halves = \(\frac{3}{2}\) which equals \(\frac{9}{6}\) — so 9 pieces of \(\frac{1}{6}\) stack up to the same length.

Figure it Out (p.172)

Q. Find equivalent fractions for the given pairs so that the fractional units are the same:
(a) \(\frac{7}{2}\) and \(\frac{3}{5}\)   (b) \(\frac{8}{3}\) and \(\frac{5}{6}\)   (c) \(\frac{3}{4}\) and \(\frac{3}{5}\)   (d) \(\frac{6}{7}\) and \(\frac{8}{5}\)
(e) \(\frac{9}{4}\) and \(\frac{5}{2}\)   (f) \(\frac{1}{10}\) and \(\frac{2}{9}\)   (g) \(\frac{8}{3}\) and \(\frac{11}{4}\)   (h) \(\frac{13}{6}\) and \(\frac{1}{9}\)
(a) \(\frac{35}{10}\) and \(\frac{6}{10}\)   (b) \(\frac{16}{6}\) and \(\frac{5}{6}\)   (c) \(\frac{15}{20}\) and \(\frac{12}{20}\)   (d) \(\frac{30}{35}\) and \(\frac{56}{35}\)
(e) \(\frac{9}{4}\) and \(\frac{10}{4}\)   (f) \(\frac{9}{90}\) and \(\frac{20}{90}\)   (g) \(\frac{32}{12}\) and \(\frac{33}{12}\)   (h) \(\frac{39}{18}\) and \(\frac{2}{18}\).

Competency-Based Questions

Scenario: Aryan reads \(\frac{2}{3}\) of his storybook on Monday and \(\frac{3}{5}\) of another storybook of the same length on Tuesday. Both books have 60 pages.
Q1. How many pages did Aryan read on Monday and Tuesday?
L3 Apply
Monday: \(\frac{2}{3}\times 60=40\) pages. Tuesday: \(\frac{3}{5}\times 60=36\) pages.
Q2. Which day did he read more, and by how many pages? Analyse using equivalent fractions.
L4 Analyse
Convert to common denom 15: \(\frac{2}{3}=\frac{10}{15}\), \(\frac{3}{5}=\frac{9}{15}\). Monday is more. Difference = 40−36 = 4 pages.
Q3. Aryan claims: "\(\frac{3}{5} > \frac{2}{3}\) because 5 > 3." Evaluate.
L5 Evaluate
Wrong. A bigger denominator alone means smaller pieces, not a bigger fraction. By converting to 15ths, \(\frac{10}{15} > \frac{9}{15}\) so \(\frac{2}{3} > \frac{3}{5}\).
Q4. Design a reading plan in which Aryan reads a different fraction each of three days such that the total for the 60-page book equals exactly 1 whole. Create your plan with fractions in lowest terms.
L6 Create
One plan: \(\frac{1}{3}+\frac{1}{4}+\frac{5}{12}=\frac{4}{12}+\frac{3}{12}+\frac{5}{12}=\frac{12}{12}=1\). Pages: 20, 15, 25.

Assertion–Reason Questions

A: \(\frac{4}{8}\) and \(\frac{1}{2}\) are equivalent.
R: Dividing numerator and denominator by the same number gives an equivalent fraction.
(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) — 4÷4 = 1, 8÷4 = 2. R explains A.
A: Since 5 > 3, \(\frac{1}{5} > \frac{1}{3}\).
R: Larger denominators mean larger pieces.
(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. \(\frac{1}{5} < \frac{1}{3}\). Larger denominators → smaller pieces.
A: To compare \(\frac{2}{3}\) and \(\frac{3}{4}\), we can use common denominator 12.
R: 12 is a common multiple of 3 and 4.
(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, and R explains why 12 works.

Frequently Asked Questions

What is an equivalent fraction?
An equivalent fraction is a fraction that names the same value as another fraction, obtained by multiplying or dividing both numerator and denominator by the same non-zero number. Example: 2/3 is equivalent to 4/6 and 6/9.
What is the simplest form of a fraction?
A fraction is in simplest form when the numerator and denominator share no common factor other than 1. Divide both by their HCF to reduce a fraction to its simplest form.
How do you compare fractions with different denominators?
Find a common denominator using the LCM of the two denominators, rewrite both fractions with that denominator, and compare the numerators. The fraction with the greater numerator is larger.
What is cross multiplication for comparing fractions?
To compare a/b and c/d, compute a times d and b times c. If a times d is greater, then a/b is greater; if smaller, then a/b is smaller; if equal, the fractions are equivalent.
Why are equivalent fractions useful?
Equivalent fractions let us add, subtract and compare fractions that have different denominators by rewriting them with a common denominator.
How do you check if 3/4 and 9/12 are equivalent?
Multiply 3 by 12 to get 36 and 4 by 9 to get 36. Since both products are equal, the fractions 3/4 and 9/12 are equivalent.
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