UPTET Paper 1 Model Test 1

Concept:Use BODMAS rule: Brackets, Of, Division, Multiplication, Addition, Subtraction. Convert mixed fractions to improper fractions. "Of" means multiplication.Explanation:Step 1: Replace mixed numbers and "of":$1\frac{1}{4} = \frac{5}{4}$, $1\frac{2}{3} = \frac{5}{3}$, $3\frac{1}{2} = \frac{7}{2}$, $2\frac{1}{6} = \frac{13}{6}$."of" becomes $\times$. The expression becomes:$\frac{1}{4} \times \frac{3}{4} \div \frac{5}{4} \times \frac{2}{5} - \left[\frac{1}{6} \div \left(\frac{3}{7} \times \frac{14}{5} \times \frac{5}{3} - \left(\frac{7}{2} - \frac{13}{6}\right)\right)\right]$.Step 2: Simplify inside the innermost bracket first:$\frac{7}{2} - \frac{13}{6} = \frac{21-13}{6} = \frac{8}{6} = \frac{4}{3}$.Step 3: Simplify $\frac{3}{7} \times \frac{14}{5} \times \frac{5}{3} = \frac{3 \times 14 \times 5}{7 \times 5 \times 3} = \frac{14}{7} = 2$.Step 4: Now inside square brackets: $2 - \frac{4}{3} = \frac{6-4}{3} = \frac{2}{3}$.So we have $\frac{1}{6} \div \frac{2}{3} = \frac{1}{6} \times \frac{3}{2} = \frac{1}{4}$.Step 5: Now the first part: $\frac{1}{4} \times \frac{3}{4} \div \frac{5}{4} \times \frac{2}{5}$.Division first: $\frac{3}{4} \div \frac{5}{4} = \frac{3}{4} \times \frac{4}{5} = \frac{3}{5}$.Then $\frac{1}{4} \times \frac{3}{5} = \frac{3}{20}$.Then $\frac{3}{20} \times \frac{2}{5} = \frac{6}{100} = \frac{3}{50}$? Wait, careful: The original expression after step 1 is $\frac{1}{4} \times \frac{3}{4} \div \frac{5}{4} \times \frac{2}{5}$. According to BODMAS, division and multiplication have equal precedence, so we evaluate left to right. That means: $\frac{1}{4} \times \frac{3}{4} = \frac{3}{16}$, then $\frac{3}{16} \div \frac{5}{4} = \frac{3}{16} \times \frac{4}{5} = \frac{12}{80} = \frac{3}{20}$, then $\frac{3}{20} \times \frac{2}{5} = \frac{6}{100} = \frac{3}{50}$. But the existing solution shows a different path: they first did $\frac{3}{4} \div \frac{5}{4} \times \frac{2}{5}$ as a group? Actually they wrote: $\frac{1}{4} \times \frac{3}{4} \div \frac{5}{4} \times \frac{2}{5}$ and then converted $\div$ to $\times$ and got $\frac{1}{2}$ later. Let's re-evaluate correctly: The original solution they gave: After converting, they wrote "⇒ $\frac{1}{4} \times \frac{3}{4} \div \frac{1}{2}$" which means they simplified $\frac{5}{4} \times \frac{2}{5}$ to $\frac{1}{2}$? But that's not correct BODMAS because they grouped division and multiplication incorrectly. The proper interpretation: The expression $\frac{1}{4} \times \frac{3}{4} \div 1\frac{1}{4} of \frac{2}{5}$ means "of" has higher precedence than division? Actually "of" is multiplication, but in the original expression, it's $\div 1\frac{1}{4} of \frac{2}{5}$. The phrase "of" binds tighter than division? In standard BODMAS, "of" is treated as multiplication but with same precedence as multiplication and division? Actually in many textbooks, "of" is considered a higher precedence than multiplication/division, essentially like a bracket. The typical rule: "of" is stronger than multiplication and division. So $a \div b \text{ of } c$ means $a \div (b \times c)$. So that is what the solution used: they treated $1\frac{1}{4} of \frac{2}{5}$ as a single term: $\frac{5}{4} \times \frac{2}{5} = \frac{1}{2}$. Then the expression becomes $\frac{1}{4} \times \frac{3}{4} \div \frac{1}{2}$. That is correct. So we need to follow that. So step-by-step: First part: $\frac{1}{4} \times \frac{3}{4} \div \left(1\frac{1}{4} \times \frac{2}{5}\right) = \frac{1}{4} \times \frac{3}{4} \div \left(\frac{5}{4} \times \frac{2}{5}\right) = \frac{1}{4} \times \frac{3}{4} \div \frac{1}{2}$.Then $\frac{3}{16} \div \frac{1}{2} = \frac{3}{16} \times 2 = \frac{3}{8}$.So the first part simplifies to $\frac{3}{8}$.Step 6: Now the whole expression: $\frac{3}{8} - \frac{1}{4} = \frac{3-2}{8} = \frac{1}{8}$.Answer:$1/8$ (Option C).