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Difficulty: MediumLinear Equations in One Variable

A bakery produces sourdough loaves using a mixture of two types of flour, Flour X and Flour Y. A standard batch requires a total of 8484 kilograms of flour. The amount of Flour X used is 1212 kilograms more than twice the amount of Flour Y used. If yy represents the amount of Flour Y, in kilograms, used in one standard batch, which of the following statements must be true? Select all such statements.

  1. yy is a positive integer multiple of 88.Answer
  2. The ratio of the amount of Flour X to the amount of Flour Y used in one batch is 5:25:2.Answer
  3. C
    Flour X accounts for 75%75\% of the total flour used in one batch.
  4. The amount of Flour X used in one batch exceeds the amount of Flour Y used by 3636 kilograms.Answer
  5. E
    The combined amount of Flour Y used in 44 standard batches is 100100 kilograms.

Answer

The correct statements are that yy is a positive integer multiple of 88, the ratio of Flour X to Flour Y used in one batch is 5:25:2, and the amount of Flour X used in one batch exceeds the amount of Flour Y used by 3636 kilograms.
Solving y+(2y+12)=84y + (2y + 12) = 84 yields y=24y = 24 kilograms of Flour Y and 6060 kilograms of Flour X. Since 2424 is divisible by 88, the statement asserting yy is a multiple of 88 is true. The ratio of Flour X to Flour Y is 60:24=5:260:24 = 5:2, making the ratio statement true. Finally, 60−24=3660 - 24 = 36, confirming that Flour X exceeds Flour Y by 3636 kilograms.

Step-by-Step Solution

1
Set up the linear equation in terms of yy.
Amount of Flour Y = yy, Amount of Flour X = 2y+122y + 12. Total amount = y+(2y+12)=84y + (2y + 12) = 84.
The problem states Flour X is 1212 kg more than twice Flour Y, and their sum equals 8484 kg.
2
Solve the linear equation for yy.
3y+12=84  ⟹  3y=72  ⟹  y=243y + 12 = 84 \implies 3y = 72 \implies y = 24.
Subtract 1212 from both sides and divide by 33 to isolate the variable yy.
3
Calculate the amount of Flour X used.
Amount of Flour X = 2(24)+12=48+12=602(24) + 12 = 48 + 12 = 60 kilograms.
Substitute y=24y = 24 back into the expression for Flour X.
4
Evaluate each given statement.
1) y=24y = 24, which is a multiple of 88 (24=8×324 = 8 \times 3). (True)
2) Ratio of Flour X to Flour Y is 60:24=5:260 : 24 = 5 : 2. (True)
3) Percentage of Flour X is 6084=57≈71.43%≠75%\frac{60}{84} = \frac{5}{7} \approx 71.43\% \neq 75\%. (False)
4) Difference is 60−24=3660 - 24 = 36 kg. (True)
5) 44 batches of Flour Y equal 4×24=964 \times 24 = 96 kg ≠100\neq 100 kg. (False)
Direct calculation confirms which statements match the derived values.

Key Concept

Formulating and solving a linear equation in one variable from a word problem scenario, followed by logical evaluation of derived quantities.
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Topics

Linear Equations in One Variable

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