There are two coaching institutions in a town, institution A and institution B, the total number of students in both institutions are 1350. The ratio of the students in institution A to that of institution B is 7: 8. There are only three classes in each Institution, Class X, Class Y and 76 Class Z. In institution A, 70% students are boys and the rest are girls, whereas in institution B the ratio of boys to girls is 11 : 7. In institution A, 4 7 of the total girls enrolled in Class Y,8/9 of the remaining enrolled in Class Z and rest in Class X. Out of the total boys in institution A, 42 6/7% enrolled in class X, 44 4/9% of the remaining in Class Y and the rest in Class Z. In Institution B, 4/11 of the total boys enrolled in class Y, and the number of boys enrolled in class Z is 5% more than the boys enrolled in class Y and rest in class X. one-fourth of the total girls are enrolled in class Z, and the number of girls enrolled in class X is 10% more than the girls enrolled in class Y.
What is the difference between the number of boys enrolled in Class Y in institution B and the number of girls enrolled in Class X in institution A?
Show Hint
- 112
- 136
- 148
- 124
The Correct Option is D
Solution and Explanation
The question asks for the difference between two specific values: the number of boys in Class Y of institution B and the number of girls in Class X of institution A.
Preliminary Calculations:
Total students = 1350.
Ratio of students in Institution A to Institution B = 7 : 8.
Number of students in Institution A = \(\frac{7}{7+8} \times 1350 = \frac{7}{15} \times 1350 = 630\).
Number of students in Institution B = \(\frac{8}{15} \times 1350 = 720\).
Institution A Analysis:
Total students = 630.
Boys = 70% of 630 = 0.70 \(\times\) 630 = 441.
Girls = 30% of 630 = 0.30 \(\times\) 630 = 189.
Girls distribution (A):
Girls in Class Y = \(\frac{4}{7} \times 189 = 108\).
Remaining girls = 189 - 108 = 81.
Girls in Class Z = \(\frac{5}{9} \times 81 = 45\).
Girls in Class X = 189 - 108 - 45 = 36.
Boys distribution (A):
Boys in Class X = \(42\frac{6}{7}% \text{ of } 441 = \frac{300}{700} \times 441 = \frac{3}{7} \times 441 = 189\).
Remaining boys = 441 - 189 = 252.
Boys in Class Y = \(44\frac{4}{9}% \text{ of } 252 = \frac{400}{900} \times 252 = \frac{4}{9} \times 252 = 112\).
Boys in Class Z = 441 - 189 - 112 = 140.
Institution B Analysis:
Total students = 720.
Ratio of boys to girls = 11 : 7.
Boys = \(\frac{11}{18} \times 720 = 440\).
Girls = \(\frac{7}{18} \times 720 = 280\).
Boys distribution (B):
Boys in Class Y = \(\frac{4}{11} \times 440 = 160\).
Boys in Class Z = 160 + 5% of 160 = 160 + 8 = 168.
Boys in Class X = 440 - 160 - 168 = 112.
Girls distribution (B):
Girls in Class Z = \(\frac{1}{4} \times 280 = 70\).
Let girls in Class Y be \(y\). Girls in Class X = \(y + 0.10y = 1.1y\).
Girls in X + Girls in Y + Girls in Z = 280 \(\implies 1.1y + y + 70 = 280\).
\(2.1y = 210 \implies y = 100\).
Girls in Class Y = 100.
Girls in Class X = 1.1 \(\times\) 100 = 110.
\hrule \vspace{0.5cm} Step 2: Detailed Explanation:
From our preliminary calculations:
Number of boys enrolled in Class Y in institution B = 160.
Number of girls enrolled in Class X in institution A = 36.
Step 3: Calculation:
Difference = (Boys in Class Y, Inst. B) - (Girls in Class X, Inst. A).
\[ \text{Difference} = 160 - 36 = 124 \] Step 4: Final Answer:
The difference is 124.
Find the total number of students enrolled in class Y of both institutions.
Show Hint
- 410
- 440
- 560
- 480
The Correct Option is D
Solution and Explanation
We need to find the total number of students (boys + girls) in Class Y from both Institution A and Institution B.
Step 2: Detailed Explanation:
First, calculate the total students in Class Y for each institution.
Institution A:
Boys in Class Y = 112.
Girls in Class Y = 108.
Total students in Class Y (A) = 112 + 108 = 220.
Institution B:
Boys in Class Y = 160.
Girls in Class Y = 100.
Total students in Class Y (B) = 160 + 100 = 260.
Step 3: Calculation:
Total students in Class Y (Both institutions) = Total in A + Total in B.
\[ \text{Total} = 220 + 260 = 480 \] Step 4: Final Answer:
The total number of students enrolled in class Y of both institutions is 480.
The number of boys enrolled in class X in institution A is what percentage more than the number of girls enrolled in class Y in institution B?
Show Hint
- 69%
- 76%
- 89%
- 56%
The Correct Option is C
Solution and Explanation
The question asks to express the difference between two values as a percentage of the second value. Specifically, how much larger is the number of boys in Class X (Inst. A) compared to the number of girls in Class Y (Inst. B).
Step 2: Key Formula or Approach:
The formula for percentage increase is:
\[ \text{Percentage More} = \frac{\text{Value 1} - \text{Value 2}}{\text{Value 2}} \times 100% \] Step 3: Detailed Explanation:
From our preliminary calculations:
Value 1: Number of boys in class X in institution A = 189.
Value 2: Number of girls in class Y in institution B = 100.
Step 4: Calculation:
Plugging the values into the formula:
\[ \text{Percentage More} = \frac{189 - 100}{100} \times 100% = \frac{89}{100} \times 100% = 89% \] Step 5: Final Answer:
The number of boys in class X in institution A is 89% more than the number of girls in class Y in institution B.
The total number of boys enrolled in classes X and Y in institution A is what percentage of the total number of girls enrolled in classes Y and Z in institution B (rounded off to the integer)?
Show Hint
- 182
- 172
- 187
- 177
The Correct Option is D
Solution and Explanation
We need to calculate a percentage. The numerator is the total number of boys in classes X and Y in institution A, and the denominator is the total number of girls in classes Y and Z in institution B.
Step 2: Key Formula or Approach:
The formula for "A is what percentage of B" is:
\[ \text{Percentage} = \frac{A}{B} \times 100% \] Step 3: Detailed Explanation:
First, calculate the required totals from our preliminary data.
Numerator (A): Total boys in classes X and Y in institution A.
Boys in Class X (A) = 189.
Boys in Class Y (A) = 112.
Total = 189 + 112 = 301.
Denominator (B): Total girls in classes Y and Z in institution B.
Girls in Class Y (B) = 100.
Girls in Class Z (B) = 70.
Total = 100 + 70 = 170.
Step 4: Calculation:
Now, calculate the percentage:
\[ \text{Percentage} = \frac{301}{170} \times 100% \approx 177.0588% \] Rounding off to the nearest integer, we get 177%.
Step 5: Final Answer:
The required percentage is 177%.
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Read the given passage and answer the questions.
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