Examkin

Question

Difficulty: Very hardTranslating and Solving Algebraic Word Problems

A commercial bakery has two bread-kneading machines, Machine X and Machine Y. Machine X can knead dough at a constant rate of pp pounds per hour. Machine Y's kneading rate is 1414 pounds per hour more than half the kneading rate of Machine X. On a busy morning, Machine X starts kneading and operates for exactly 55 hours. Machine Y starts operating 11 hour after Machine X starts and operates for the next 44 hours. Together, the two machines knead a total of 336336 pounds of dough. What is the total number of pounds of dough kneaded by Machine Y?

  1. A
    34
  2. B
    116
  3. 136Answer
  4. D
    148
  5. E
    200

Answer

The total number of pounds of dough kneaded by Machine Y is 136.
To find the total dough kneaded by Machine Y, we first define the rates of both machines. Let Machine X's rate be pp pounds per hour. Machine Y's rate is 1414 more than half of Machine X's rate, which is written as 0.5p+140.5p + 14 pounds per hour. Machine X runs for 55 hours, producing 5p5p pounds of dough. Machine Y runs for 44 hours, producing 4(0.5p+14)=2p+564(0.5p + 14) = 2p + 56 pounds of dough. Setting their sum equal to the total of 336336 gives the equation 7p+56=3367p + 56 = 336, which simplifies to 7p=2807p = 280 and yields p=40p = 40. Machine Y's total work is then 4(0.5(40)+14)=1364(0.5(40) + 14) = 136 pounds.

Step-by-Step Solution

1
Define the variable for Machine X's rate and write the expression for Machine Y's rate based on the text.
Machine X's rate is pp pounds per hour. Machine Y's rate is 0.5p+140.5p + 14 pounds per hour.
We need to express both rates in terms of a single variable to set up the equation.
2
Determine the operating duration for each machine.
Machine X operates for 55 hours. Machine Y operates for 44 hours.
Machine X starts first and runs for 55 hours. Machine Y starts 11 hour later and operates for the remaining 44 hours.
3
Write the equation for the total pounds of dough kneaded by both machines combined.
5p+4(0.5p+14)=3365p + 4(0.5p + 14) = 336
Total work is the sum of the work done by Machine X (rate times time) and Machine Y (rate times time).
4
Solve the equation for the variable pp.
5p+2p+56=336  ⟹  7p+56=336  ⟹  7p=280  ⟹  p=405p + 2p + 56 = 336 \implies 7p + 56 = 336 \implies 7p = 280 \implies p = 40.
Distribute the 44 and combine like terms to isolate pp.
5
Calculate the total work done specifically by Machine Y.
Machine Y's total work = 4(0.5(40)+14)=4(20+14)=4(34)=1364(0.5(40) + 14) = 4(20 + 14) = 4(34) = 136 pounds.
The question asks for the total pounds of dough kneaded by Machine Y, which is its rate multiplied by its operating time.

Key Concept

Translating verbal descriptions of rates and times into algebraic equations and solving them.

Alternative Method

Instead of solving for pp first, we can write the equation directly in terms of Machine Y's work. Let yy be the work done by Machine Y. Since Machine Y worked for 44 hours, its rate is y4\frac{y}{4}. This rate is 1414 more than half of Machine X's rate, so y4=0.5RX+14  ⟹  RX=2(y4−14)=y2−28\frac{y}{4} = 0.5R_X + 14 \implies R_X = 2(\frac{y}{4} - 14) = \frac{y}{2} - 28. Machine X's work is 5RX=5(y2−28)=2.5y−1405 R_X = 5(\frac{y}{2} - 28) = 2.5y - 140. Since the total work is 336336, we have (2.5y−140)+y=336  ⟹  3.5y=476  ⟹  y=136(2.5y - 140) + y = 336 \implies 3.5y = 476 \implies y = 136.
Estimated Time:3m 0s
Rate this question

Topics

Translating and Solving Algebraic Word Problems

More questions from this topic