This MCQ module is based on: Measuring Lengths with Fractions
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Measuring Lengths with Fractions
🎓 Class 6
Mathematics
CBSE
Theory
Ch 7 — Fractions
⏱ ~35 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Measuring Lengths with Fractions
Targeting Class 6 level in Fractions, with Basic difficulty.
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7.3 Measuring Using Fractional Units
How many \(\frac{1}{2}\) pieces make 1 whole? Two. How many \(\frac{1}{3}\) pieces make 1 whole? Three. In general, \(n\) copies of \(\frac{1}{n}\) make exactly 1 whole.
If a pole is as long as 5 pieces of \(\frac{1}{4}\) m each, then its length is \(5 \times \frac{1}{4} = \frac{5}{4}\) m. When we put together several copies of a unit fraction, we build a new fraction.
Definition
A fraction? \(\frac{a}{b}\) means \(a\) copies of the fractional unit \(\frac{1}{b}\). The bottom number \(b\) is the denominator (how many equal parts the whole is split into). The top number \(a\) is the numerator (how many of those parts we take).
7.4 Marking Fractions on the Number Line
A number line is a powerful tool for visualising fractions. To mark \(\frac{1}{3}, \frac{2}{3}, \frac{3}{3}=1\), we divide the unit segment from 0 to 1 into 3 equal parts.
Marking thirds on the number line.
To mark a fraction like \(\frac{5}{4}\) on the number line, we split each unit from 0 to 1, 1 to 2, etc., into 4 equal parts. Then we count 5 quarter-jumps from 0 — landing one quarter past 1.
\(\frac{5}{4}\) is one quarter past 1 on the number line.
In-text: How many fractions lie between 0 and 1?
Answer: Uncountably many — between any two fractions you can always find another.
Answer: Uncountably many — between any two fractions you can always find another.
Figure it Out (p.160)
Q1. On a number line, draw lines of length \(\frac{1}{10}, \frac{3}{10},\) and \(\frac{4}{5}\).
Divide the unit into 10 equal parts. \(\frac{1}{10}\) is the first tick, \(\frac{3}{10}\) is the third tick, and \(\frac{4}{5}=\frac{8}{10}\) is the eighth tick.
Q2. Write five more fractions of your choice and mark them on the number line.
Example: \(\frac{1}{4}, \frac{1}{2}, \frac{2}{3}, \frac{5}{6}, \frac{7}{8}\) — mark each by dividing the unit into the denominator parts.
Q3. How many fractions lie between 0 and 1? Discuss.
An uncountably infinite number. Between any two fractions (say \(\frac{1}{2}\) and \(\frac{1}{3}\)) another one always exists, e.g., \(\frac{5}{12}\).
Q4. The blue line runs from 0 to \(\frac{3}{2}\) where each unit is divided into 2 equal parts. Write the fraction for the black line that ends at \(\frac{3}{2}\) unit.
Length = \(\frac{3}{2}\) units.
7.5 Mixed Fractions
Fractions like \(\frac{7}{2}, \frac{4}{3}, \frac{7}{3}\) are called improper fractions? because the numerator is bigger than the denominator — they are more than 1 whole.
We can split such a fraction into a whole part + proper fraction. For example: \[\frac{7}{2} = \frac{6}{2} + \frac{1}{2} = 3 + \frac{1}{2} = 3\tfrac{1}{2}\] This is called a mixed fraction.
Example
How many whole units are in \(\frac{7}{2}\)? → 3 wholes, with \(\frac{1}{2}\) left over. So \(\frac{7}{2}=3\tfrac{1}{2}\).How many wholes in \(\frac{4}{3}\)? → 1 whole and \(\frac{1}{3}\). So \(\frac{4}{3}=1\tfrac{1}{3}\).
How many wholes in \(\frac{7}{3}\)? → 2 wholes and \(\frac{1}{3}\). So \(\frac{7}{3}=2\tfrac{1}{3}\).
Activity: Strip Ruler for Mixed Fractions
L3 ApplyMaterials: A long paper strip (at least 30 cm), ruler, pencil.
Predict: Where on a 30 cm strip will the fraction \(\frac{9}{4}\) (in units of 10 cm) land?
- Mark 0, 1, 2, 3 at every 10 cm on the strip.
- Divide each 10 cm unit into 4 equal parts (2.5 cm each).
- Count 9 small parts from 0 — label the point \(\frac{9}{4}\).
- Rewrite \(\frac{9}{4}\) as a mixed fraction and check it matches the position.
\(\frac{9}{4}=2\tfrac{1}{4}\). Two full units (20 cm) plus one quarter (2.5 cm) = 22.5 cm from 0 — exactly where the 9th tick lands.
Figure it Out (p.162)
Q1. How many whole units are there in \(\frac{7}{2}\)?
3 wholes.
Q2. How many whole units are there in \(\frac{4}{3}\) and in \(\frac{7}{3}\)?
1 whole in \(\frac{4}{3}\); 2 wholes in \(\frac{7}{3}\).
Competency-Based Questions
Scenario: A carpenter has a long bamboo pole 3 m in length. She needs pieces of length \(\frac{1}{4}\) m each to build the legs of a small stool.
Q1. How many \(\frac{1}{4}\) m pieces can she cut from the pole?
L3 Apply3 m = \(\frac{12}{4}\) m → 12 pieces of \(\frac{1}{4}\) m each.
Q2. If she cut 9 pieces and used them, analyse how much pole (in mixed-fraction form) remains.
L4 AnalyseUsed = \(\frac{9}{4}\) m; remaining = \(3 - \frac{9}{4} = \frac{12-9}{4}=\frac{3}{4}\) m. As a mixed fraction \(\frac{3}{4}\) is already proper.
Q3. Her assistant says: "Since each piece is tiny, we can cut 13 pieces from 3 m." Evaluate his claim.
L5 EvaluateWrong. 13 pieces need \(13 \times \frac{1}{4} = \frac{13}{4}=3\tfrac{1}{4}\) m which is more than 3 m — impossible.
Q4. Design a stool plan using at most 10 pieces of \(\frac{1}{4}\) m and at most 4 pieces of \(\frac{1}{2}\) m, totalling ≤ 3 m. Write your plan as a sum of fractions.
L6 CreateOne plan: 8 × \(\frac{1}{4}\) + 2 × \(\frac{1}{2}\) = 2 + 1 = 3 m exactly. Many valid plans possible.
Assertion–Reason Questions
A: \(\frac{7}{2}=3\tfrac{1}{2}\).
R: An improper fraction can always be written as whole + proper fraction.
R: An improper fraction can always be written as whole + proper fraction.
(a) — \(\frac{7}{2}=\frac{6}{2}+\frac{1}{2}=3+\frac{1}{2}\). R explains A.
A: Between 0 and 1 there are exactly 99 fractions.
R: Only fractions with denominator 100 exist between 0 and 1.
R: Only fractions with denominator 100 exist between 0 and 1.
Both false — there are uncountably many fractions between 0 and 1, and fractions exist with every positive denominator.
A: On the number line, \(\frac{3}{4}\) lies between 0 and 1.
R: Any proper fraction \(\frac{a}{b}\) with \(a < b\) is less than 1.
R: Any proper fraction \(\frac{a}{b}\) with \(a < b\) is less than 1.
(a) — Both true and R explains A.
Frequently Asked Questions
How do we show fractions on a number line?
Divide each unit segment on the number line into the same number of equal parts as the denominator. Each mark is one unit fraction. To plot 3/4, start at 0 and move 3 of the 4 equal parts between 0 and 1.
What is a mixed fraction?
A mixed fraction has a whole-number part and a fractional part, like 2 1/3. It means two whole units plus one-third of another unit.
How do you convert an improper fraction to a mixed fraction?
Divide the numerator by the denominator. The quotient is the whole number and the remainder over the denominator is the fractional part. For 7/3, 7 divided by 3 is 2 remainder 1, so 7/3 equals 2 1/3.
Why does a ruler use 10 small lines between centimetres?
Each centimetre is divided into 10 equal millimetres, so every small line is 1/10 cm. This lets us measure lengths like 4.7 cm as 4 and 7/10 cm.
Can the same length be written as different fractions?
Yes. 2/4 of a metre and 1/2 of a metre describe the same length. These are equivalent fractions that represent the same measurement using different fractional units.
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