Direction: Study the information given below and answer the question that follows.
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Direction: Study the information given below and answer the question that follows.
Three online homestay reservation websites X, Y and Z have 3 star, 4 star and 5 star homestays listed on each of their websites. The details of number of different category homestays given below:
- One homestay can be listed on exactly one website.
- The total number of homestays listed on all three websites taken together is 360.
- The total number of 4 star homestays is twice the total number of 3 star homestays on all the three websites taken together.
- On website X, out of total 120 homestays, 85 homestays are 5 star homestays.
- On website Y, half of the total listed homestays are 5 star homestays. Also the number of 3 star homestays is the same as the number of 4 star homestays listed on website Y.
- The number of 3 star homestays listed on website Z is equal to the number of 3 star homestays listed on website X.
- The difference between the number of 3 star homestays listed on website Z and the number of 4 star homestays listed on website X is 15.
Question 1: What is the total number of homestays listed on website Y?
Show answer & explanation
By setting up a system of linear equations based on the constraints provided, we find that the number of 3-star homestays on website X is 10, which satisfies all conditions including the total count of 360 homestays and the 2:1 ratio of 4-star to 3-star homestays.
Step-by-step Derivation:
Step 1: Define variables. Let X3, X4, X5 be homestays on website X; Y3, Y4, Y5 on Y; and Z3, Z4, Z5 on Z.
Step 2: From the data: Total = 360. X_total = 120. X5 = 85. Thus, X3 + X4 = 120 - 85 = 35.
Step 3: Website Y: Y5 = 0.5 * Y_total. Since Y3 = Y4, then Y3 + Y4 = 0.5 * Y_total. Thus, Y3 = Y4 = 0.25 * Y_total.
Step 4: Website Z: Z3 = X3. Given |Z3 - X4| = 15. Since X3 + X4 = 35, we substitute X4 = 35 - X3. So, |X3 - (35 - X3)| = 15 => |2X3 - 35| = 15.
Step 5: Solve for X3: Either 2X3 - 35 = 15 (X3 = 25) or 2X3 - 35 = -15 (X3 = 10).
Step 6: Test X3 = 10: X3 = 10, X4 = 25, Z3 = 10. Total 3-star (T3) = X3 + Y3 + Z3 = 10 + Y3 + 10 = 20 + Y3. Total 4-star (T4) = X4 + Y4 + Z4 = 25 + Y3 + Z4.
Step 7: Use T4 = 2 * T3: 25 + Y3 + Z4 = 2 * (20 + Y3) => 25 + Y3 + Z4 = 40 + 2Y3 => Z4 = 15 + Y3.
Step 8: Use Total = 360: X_total + Y_total + Z_total = 360 => 120 + (4Y3) + (Z3 + Z4 + Z5) = 360 => 120 + 4Y3 + 10 + (15 + Y3) + Z5 = 360 => 5Y3 + Z5 = 225.
Step 9: If X3 = 25, then X4 = 10, Z3 = 25. T3 = 50 + Y3, T4 = 10 + Y3 + Z4. T4 = 2T3 => 10 + Y3 + Z4 = 100 + 2Y3 => Z4 = 90 + Y3. Total = 120 + 4Y3 + 25 + 90 + Y3 + Z5 = 360 => 5Y3 + Z5 = 125. Both X3=10 and X3=25 are mathematically possible, but looking at the provided options for Question 2, only 10 is listed as an option (Option B). Checking Question 1: If X3=10, Y_total = 4Y3. If Y3=30, Y_total=120. If Y3=60, Y_total=240. Given the options, X3=10 is the intended answer for Question 2.