Multiple Choice Type
Q1: Mercy solved \(\frac{2}{7}\) part of an exercise while John solved \(\frac{4}{5}\) of it, who solved lesser part?
Step 1: Compare the two fractions: \( \frac{2}{7} \) and \( \frac{4}{5} \)
Step 2: Convert both fractions to have a common denominator.
LCM of 7 and 5 = 35
\( \frac{2}{7} = \frac{2 \times 5}{7 \times 5} = \frac{10}{35} \)
\( \frac{4}{5} = \frac{4 \times 7}{5 \times 7} = \frac{28}{35} \)
Step 3: Now compare: \( \frac{10}{35} \lt \frac{28}{35} \)
So, Mercy solved the lesser part.
Correct Answer: i. Mercy
Q2: Fractions \(\frac{7}{8},\ \frac{17}{12}\ and\ \frac{41}{48}\) in the form of their equivalent fractions are:
Option 1: HCF of 315 and 360 is 45.
\(\frac{315}{360}\ =\ \frac{315\div45}{360\div45}=\frac{7}{8}\)
HCF of 510 and 360 is 30.
\(\frac{510}{360}\ =\ \frac{510\div30}{360\div30}=\frac{17}{12}\)
HCF of 126 and 360 is 18.
\(\frac{126}{360}\ =\ \frac{126\div18}{360\div18}=\frac{7}{20}\)
Since \(\frac{126}{360}\) doesnot match with \(\frac{41}{48}\) so it is incorrect option.
Option 2: HCF of 315 and 180 is 45.
\(\frac{315}{180}\ =\ \frac{315\div45}{180\div45}=\frac{7}{30}\)
Since \(\frac{315}{180}\) doesnot match with \(\frac{7}{8}\) so it is incorrect option.
Option 3: Since \(\frac{7}{360}\) doesnot match with \(\frac{7}{8}\) so it is incorrect option.
Correct Answer: iv. none of these
Q3: The difference between the fractions \(\frac{3}{5}\ and\ \frac{7}{10}\) is:
Step 1: Convert both fractions to the same denominator.
LCM of 5 and 10 = 10
\( \frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \)
So now we compare:
\( \frac{7}{10} – \frac{6}{10} = \frac{7 – 6}{10} = \frac{1}{10} \)
Answer: ii. \( \frac{1}{10} \)
Q4: \(2\frac{1}{3}+4\frac{1}{3}-3\frac{1}{3}\) is equal to
Step 1: Convert mixed numbers to improper fractions.
\( 2\frac{1}{3} = \frac{(3 \times 2 + 1)}{3} = \frac{7}{3} \)
\( 4\frac{1}{3} = \frac{(3 \times 4 + 1)}{3} = \frac{13}{3} \)
\( 3\frac{1}{3} = \frac{(3 \times 3 + 1)}{3} = \frac{10}{3} \)
Step 2: Perform the operations:
\( \frac{7}{3} + \frac{13}{3} – \frac{10}{3} \)
All fractions have the same denominator, so we can combine the numerators:
\( \frac{(7 + 13 – 10)}{3} = \frac{10}{3} \)
Step 3: Convert the result back to a mixed number:
\( \frac{10}{3} = 3\frac{1}{3} \)
Correct Answer: i. \( 3\frac{1}{3} \)
Q5: Which of the following is greater?
\(\frac{2}{7}\ of\ \frac{3}{4}\ or\ \frac{3}{5}\ of\ \frac{5}{8}\)
Step 1: Calculate each expression.
First Expression:
\( \frac{2}{7} \times \frac{3}{4} = \frac{6}{28} = \frac{3}{14} \)
Second Expression:
\( \frac{3}{5} \times \frac{5}{8} = \frac{15}{40} = \frac{3}{8} \)
Step 2: Compare \( \frac{3}{14} \) and \( \frac{3}{8} \)
Find LCM of denominators 14 and 8 = 56
\( \frac{3}{14} = \frac{3 \times 4}{14 \times 4} = \frac{12}{56} \)
\( \frac{3}{8} = \frac{3 \times 7}{8 \times 7} = \frac{21}{56} \)
Clearly, \( \frac{12}{56} \lt \frac{21}{56} \)
Correct Answer: ii. \( \frac{3}{5} \) of \( \frac{5}{8} \)
Q6: By what number should \(7\frac{1}{3}\) be multiplied to get \(5\frac{2}{3}\)?
Step 1: Convert the mixed numbers to improper fractions.
\( 7\frac{1}{3} = \frac{(7 \times 3 + 1)}{3} = \frac{22}{3} \)
\( 5\frac{2}{3} = \frac{(5 \times 3 + 2)}{3} = \frac{17}{3} \)
Step 2: Let the required number be \( x \).
So, \( \frac{22}{3} \times x = \frac{17}{3} \)
Step 3: Solve for \( x \):
\( x = \frac{\frac{17}{3}}{\frac{22}{3}} = \frac{17}{3} \div \frac{22}{3} = \frac{17}{3} \times \frac{3}{22} = \frac{17}{22} \)
Correct Answer: iii. \( \frac{17}{22} \)
Q7: The cost of \(3\frac{3}{4}\) kg apples is ₹ 600; the rate of apple per kg is:
Step 1: Convert the mixed number to improper fraction.
\( 3\frac{3}{4} = \frac{(3 \times 4 + 3)}{4} = \frac{15}{4} \) kg
Step 2: Use the formula:
Rate per kg = Total Cost ÷ Total Weight
₹600 ÷ \( \frac{15}{4} \) = ₹600 × \( \frac{4}{15} \)
= ₹\( \frac{600 \times 4}{15} = \frac{2400}{15} = 160 \)
Correct Answer: iii. ₹160
Q8: \(\frac{1}{3}\) part of a work is done in one hours. The part of work done in \(2\frac{1}{5}\) hours is:
Step 1: Work done in 1 hour = \( \frac{1}{3} \)
So, work done in more time will be more.
Step 2: Work done in \( 2\frac{1}{5} \) hours = \( \frac{1}{3} \times 2\frac{1}{5} \)
Step 3: Convert mixed number to improper fraction:
\( 2\frac{1}{5} = \frac{(2 \times 5 + 1)}{5} = \frac{11}{5} \)
Now multiply:
\( \frac{1}{3} \times \frac{11}{5} = \frac{11}{15} \)
Correct Answer: iv. \( \frac{1}{3} \times 2\frac{1}{5} \)
Q9: A rectangular sheet of paper is \(11\frac{1}{2}\) cm long and \(8\frac{1}{2}\) cm wide. Its perimeter is:
Step 1: Formula for the perimeter of a rectangle is:
Perimeter = 2 × (Length + Width)
Step 2: Convert the mixed numbers to improper fractions:
\( 11\frac{1}{2} = \frac{(11 \times 2 + 1)}{2} = \frac{23}{2} \) cm (length)
\( 8\frac{1}{2} = \frac{(8 \times 2 + 1)}{2} = \frac{17}{2} \) cm (width)
Step 3: Now, substitute into the perimeter formula:
Perimeter = \( 2 \times \left( \frac{23}{2} + \frac{17}{2} \right) = 2 \times \frac{40}{2} = 2 \times 20 = 40 \) cm
Correct Answer: ii. 40 cm
Q10: Peter can read a novel in 17 days. If he devotes \(2\frac{3}{7}\) hours per day, in how many hours (in all) he read the whole novel:
Step 1: The total time Peter spends reading the novel is the product of the number of days and hours he reads each day.
\( \text{Total Hours} = 17 \times 2\frac{3}{7} \) hours
Step 2: Convert the mixed number to an improper fraction:
\( 2\frac{3}{7} = \frac{(2 \times 7 + 3)}{7} = \frac{17}{7} \)
Now, calculate:
\( 17 \times \frac{17}{7} = \frac{17 \times 17}{7} = \frac{289}{7} \)
\( \frac{289}{7} = 41\frac{2}{7} \) hours
Correct Answer: iii. \( \left( 2\frac{1}{7} \times 17 \right) \) hours
Q11: \(\frac{1}{3}-\left[\frac{1}{3}-\left\{\frac{1}{3}-\left(\frac{1}{3}-\overline{\frac{1}{3}-\frac{1}{3}}\right)\right\}\right]\) is equal to:
Step 1: Begin with the innermost bracket: \( \overline{\frac{1}{3} – \frac{1}{3}} \).
\( \frac{1}{3} – \frac{1}{3} = 0 \)
Now, substitute this result into the next bracket:
\( \frac{1}{3} – 0 = \frac{1}{3} \)
Now, substitute this into the next outer bracket:
\( \frac{1}{3} – \frac{1}{3} = 0 \)
Finally, substitute into the outermost expression:
\( \frac{1}{3} – 0 = \frac{1}{3} \)
So the final result is \( 0 \).
Correct Answer: iii. 0
Q12: \(\frac{1}{5}\div\frac{3}{7}\times\frac{5}{4}\) is equal to:
Step 1: First, let’s simplify the division and multiplication operations.
Division of fractions is equivalent to multiplying by the reciprocal.
\( \frac{1}{5} \div \frac{3}{7} = \frac{1}{5} \times \frac{7}{3} \)
Step 2: Now, multiply this result by \( \frac{5}{4} \):
\( \frac{1}{5} \times \frac{7}{3} \times \frac{5}{4} \)
This is the same as the second option.
Correct Answer: ii. \( \frac{1}{5} \times \frac{7}{3} \times \frac{5}{4} \)
Q13: \(\left\{\frac{3}{5}\div\left(\frac{3}{5}+\frac{2}{5}\right)\right\}\div\frac{3}{5}\) is equal to:
Step 1: Start with the addition inside the parentheses:
\( \frac{3}{5} + \frac{2}{5} = \frac{3 + 2}{5} = \frac{5}{5} = 1 \)
Step 2: Now, divide \( \frac{3}{5} \) by the result (which is 1):
\( \frac{3}{5} \div 1 = \frac{3}{5} \)
Step 3: Now, divide this result by \( \frac{3}{5} \):
\( \frac{3}{5} \div \frac{3}{5} = 1 \)
Correct Answer: iv. 1
Q14: What fraction of an hour is 40 minutes?
Step 1: We know that 1 hour is 60 minutes.
So, to find what fraction of an hour 40 minutes represents, we divide 40 by 60.
\( \frac{40}{60} = \frac{2}{3} \)
Correct Answer: ii. \( \frac{40}{60} \)
Q15: \(\frac{3}{11}\) of Shyam’s salary is ₹ 7,260, then his salary is:
Step 1: Let Shyam’s salary be \( x \). According to the problem:
\( \frac{3}{11} \times x = 7260 \)
Step 2: Solve for \( x \):
\( x = \frac{7260 \times 11}{3} \)
\( x = \frac{79860}{3} = 26620 \)
Correct Answer: iv. ₹26620
Q16: The number that should divide \(2\frac{1}{3}\) to get 1 is:
Step 1: Let the required number be \( x \). According to the problem:
\( \frac{2\frac{1}{3}}{x} = 1 \)
Step 2: Convert \( 2\frac{1}{3} \) to an improper fraction:
\( 2\frac{1}{3} = \frac{(2 \times 3 + 1)}{3} = \frac{7}{3} \)
Now, we have:
\( \frac{\frac{7}{3}}{x} = 1 \)
Step 3: Multiply both sides by \( x \) and solve for \( x \):
\( \frac{7}{3} = x \)
Therefore, the required number is \( \frac{7}{3} = 2\frac{1}{3} \).
Correct Answer: iii. \( 2\frac{1}{3} \)
Q17: Statement 1: \(\frac{3}{5}\) is a fraction between \(\frac{1}{2}\ and\ \frac{2}{3}\).
Statement 2 : If \(\frac{a}{b}\ and\ \frac{c}{d}\) are two given fractions, then \(\frac{a+b}{b+d}\) lies between \(\frac{a}{b}\ and\ \frac{c}{d}\).
Which of the following options is correct?
Step 1: Check Statement 1:
Let us compare \( \frac{3}{5} \) with \( \frac{1}{2} \) and \( \frac{2}{3} \):
Convert all to decimal:
\( \frac{1}{2} = 0.5 \), \( \frac{3}{5} = 0.6 \), \( \frac{2}{3} \approx 0.6667 \)
Clearly,
\( \frac{1}{2} < \frac{3}{5} < \frac{2}{3} \)
So, Statement 1 is true.
Step 2: Check Statement 2:
The formula:
If \( \frac{a}{b} < \frac{c}{d} \), then \( \frac{a+c}{b+d} \) lies between them.
This is known as the Mediant Property of rational numbers.
Hence, Statement 2 is also true.
Correct Answer: i. Both the statements are true.
Q18: Statement 1: \(\frac{7}{5},\ \frac{2}{15},\ \frac{5}{2},\ \frac{1}{3}\) in descending order of magnitude is \(\frac{7}{5}>\frac{5}{2}>\frac{2}{15}>\frac{1}{3}\).
Statement 2: After converting the given fractions into like fractions, the fraction with greater numerator is greater.
Which of the following options is correct?
Step 1: Let’s evaluate Statement 1:
We need to check if the fractions \( \frac{7}{5}, \frac{2}{15}, \frac{5}{2}, \frac{1}{3} \) are in the descending order as stated.
Convert the fractions into decimal form:
\( \frac{7}{5} = 1.4, \frac{2}{15} = 0.1333, \frac{5}{2} = 2.5, \frac{1}{3} = 0.3333 \)
Now compare the values: \( 2.5 > 1.4 > 0.3333 > 0.1333 \).
So, Statement 1 is false because the correct order should be: \( \frac{5}{2} > \frac{7}{5} > \frac{1}{3} > \frac{2}{15} \).
Step 2: Let’s evaluate Statement 2:
When converting fractions to like fractions (same denominator), the fraction with the greater numerator will indeed be greater.
So, Statement 2 is true.
Correct Answer: iv. Statement 1 is false, and Statement 2 is true.
Q19: Assertion (A): Let us consider the product of a proper fraction \(\frac{3}{4}\) and an improper fraction \(1\frac{1}{5}\). The product is \(\frac{9}{10}\). Then \(\frac{3}{4} < \frac{9}{10} < 1\frac{1}{5}\).
Reason (R): The product of a proper fraction and an improper fraction is less than the proper fraction but greater than the improper fraction.
Step 1: Let’s evaluate the assertion (A):
We are given \( \frac{3}{4} \) (a proper fraction) and \( 1\frac{1}{5} \) (an improper fraction). First, convert \( 1\frac{1}{5} \) into an improper fraction:
\( 1\frac{1}{5} = \frac{6}{5} \).
Now, multiply the two fractions:
\( \frac{3}{4} \times \frac{6}{5} = \frac{18}{20} = \frac{9}{10} \).
This confirms the product is \( \frac{9}{10} \), and we need to check if \( \frac{3}{4} < \frac{9}{10} < 1\frac{1}{5} \).
Convert \( \frac{3}{4} = 0.75 \), \( \frac{9}{10} = 0.9 \), and \( 1\frac{1}{5} = 1.2 \).
Clearly, \( 0.75 < 0.9 < 1.2 \), so Assertion (A) is true.
Step 2: Let’s evaluate the reason (R):
The reason claims that the product of a proper fraction and an improper fraction is less than the proper fraction but greater than the improper fraction. This is incorrect. The product of a proper fraction and an improper fraction is generally smaller than both the proper and the improper fractions.
Thus, Reason (R) is false.
Correct Answer: i. A is true, R is false.
Q20: Assertion (A): The product of two improper fractions say, \(1\frac{1}{3}\ and\ 2\frac{1}{2}\) is greater than both \(1\frac{1}{3}\ and\ 2\frac{1}{2}\).
Reason (R): The product of two improper positive fractions is greater than each of the improper fractions multiplied together.
Step 1: Let’s evaluate Assertion (A):
We are given the improper fractions \( 1\frac{1}{3} \) and \( 2\frac{1}{2} \). First, convert these mixed fractions to improper fractions:
\( 1\frac{1}{3} = \frac{4}{3} \), \( 2\frac{1}{2} = \frac{5}{2} \).
Now, multiply the two fractions:
\( \frac{4}{3} \times \frac{5}{2} = \frac{20}{6} = \frac{10}{3} \).
The product \( \frac{10}{3} = 3.3333 \). Now compare this with \( 1\frac{1}{3} = 1.3333 \) and \( 2\frac{1}{2} = 2.5 \).
Clearly, \( \frac{10}{3} = 3.3333 \) is greater than both \( 1\frac{1}{3} \) and \( 2\frac{1}{2} \).
So, Assertion (A) is true.
Step 2: Let’s evaluate Reason (R):
Reason (R) states that the product of two improper positive fractions is greater than each of the improper fractions multiplied together.
This is correct because the product of two positive fractions, whether proper or improper, is typically smaller than the fractions themselves. For example, \( \frac{4}{3} \times \frac{5}{2} = \frac{10}{3} \) is indeed greater than \( \frac{4}{3} \) and \( \frac{5}{2} \).
So, Reason (R) is true.
Correct Answer: iii. Both A and R are true.
Q21: Assertion (A): The fraction \(\frac{32}{56}\) can be reduced to \(\frac{4}{7}\).
Reason (R): Two fractions \(\frac{p}{q}\ and\ \frac{r}{s}\) are said to be equivalent only when ps = rq.
Step 1: Let’s evaluate Assertion (A):
We are given \( \frac{32}{56} \). To reduce the fraction, we divide both the numerator and denominator by their greatest common divisor (GCD), which is 8:
\( \frac{32}{56} = \frac{32 \div 8}{56 \div 8} = \frac{4}{7} \).
Thus, Assertion (A) is true.
Step 2: Now, let’s evaluate Reason (R):
The condition \( ps = rq \) is the definition of equivalent fractions. If two fractions are equivalent, the cross-products of the numerators and denominators are equal.
So, Reason (R) is true.
Correct Answer: iii. Both A and R are true.
