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Zorluk: OrtaTranslating and Solving Algebraic Word Problems

A manufacturing plant operates two machines, Machine X and Machine Y, to produce identical components. Machine Y produces 12 fewer components per hour than Machine X. On a certain day, Machine X operated for 5 hours and Machine Y operated for 7 hours. If the two machines produced a combined total of 636 components, how many components did Machine X produce on that day?

  1. A
    46
  2. B
    60
  3. C
    230
  4. D
    270
  5. 300Cevap

Cevap

300
To find the number of components Machine X produced, let xx represent Machine X's hourly production rate. This makes Machine Y's hourly rate x−12x - 12. Since Machine X operated for 5 hours and Machine Y operated for 7 hours to produce a total of 636 components, we can write the equation: 5x+7(x−12)=6365x + 7(x - 12) = 636. Distributing the 7 yields 5x+7x−84=6365x + 7x - 84 = 636, which simplifies to 12x−84=63612x - 84 = 636. Adding 84 to both sides gives 12x=72012x = 720, and dividing by 12 yields x=60x = 60 components per hour for Machine X. To find the total produced by Machine X, we multiply its hourly rate by the 5 hours it operated: 5×60=3005 \times 60 = 300. This matches the correct option.

Adım Adım Çözüm

1
Define variables for the hourly rates of both machines.
Let xx be the number of components Machine X produces per hour. Since Machine Y produces 12 fewer components per hour, Machine Y's hourly rate is x−12x - 12.
Establishing algebraic expressions for each machine's rate allows us to set up a linear equation.
2
Set up an equation representing the total combined production.
The total production is the sum of the components produced by Machine X in 5 hours and Machine Y in 7 hours: 5x+7(x−12)=6365x + 7(x - 12) = 636.
The sum of the products of each machine's rate and its operating time equals the total combined production of 636 components.
3
Solve the equation for the hourly rate of Machine X, xx.
Distribute the 7: 5x+7x−84=6365x + 7x - 84 = 636. Combine like terms: 12x−84=63612x - 84 = 636. Add 84 to both sides: 12x=72012x = 720. Divide by 12: x=60x = 60.
Solving for xx gives the hourly rate of Machine X.
4
Calculate the total components produced by Machine X.
Multiply the hourly rate of Machine X by its hours of operation: 5 hours×60 components/hour=3005 \text{ hours} \times 60 \text{ components/hour} = 300.
The question asks for the total components produced by Machine X, not its hourly rate.

Anahtar Kavram

Setting up and solving single-variable linear equations from verbal descriptions

Alternatif Yöntem

Instead of defining the variable as Machine X's rate, we could define yy as Machine Y's hourly rate. Then Machine X's rate is y+12y + 12. The total equation becomes 5(y+12)+7y=636  ⟹  5y+60+7y=636  ⟹  12y=576  ⟹  y=485(y + 12) + 7y = 636 \implies 5y + 60 + 7y = 636 \implies 12y = 576 \implies y = 48. Since Machine X's hourly rate is y+12=60y + 12 = 60, it produced 5×60=3005 \times 60 = 300 components.
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Translating and Solving Algebraic Word Problems

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