Question

1. machine a produces twice as many gadgets as machine b. machine c produces 50 more than machine b. if at the end of the day 6170 gadgets are produced, how many does each machine produce?
2. two ships are 120 nautical miles apart, and they are traveling toward one another. the rate of the one ship is 4 knots greater than the rate of the other. find the rate of each ship if it takes them 3 hr to meet.

Explanation:

Problem 39

Step1: Define variables

Let the number of gadgets produced by Machine B be $x$. Then Machine A produces $2x$ gadgets, and Machine C produces $x + 50$ gadgets.

Step2: Set up the equation

The total number of gadgets produced is the sum of the gadgets produced by each machine, so $2x + x + (x + 50) = 6170$.

Step3: Simplify and solve the equation

Combine like terms: $4x + 50 = 6170$. Subtract 50 from both sides: $4x = 6170 - 50 = 6120$. Divide both sides by 4: $x = \frac{6120}{4} = 1530$.

Step4: Find the number of gadgets for each machine

Machine B: $x = 1530$. Machine A: $2x = 2\times1530 = 3060$. Machine C: $x + 50 = 1530 + 50 = 1580$.

Step1: Define variables

Let the rate of the slower ship be $x$ knots. Then the rate of the faster ship is $x + 4$ knots.

Step2: Use the distance formula

The distance traveled by each ship in 3 hours is $d = rt$ (distance = rate × time). The sum of their distances is 120 nautical miles. So, $3x + 3(x + 4) = 120$.

Step3: Simplify and solve the equation

Distribute: $3x + 3x + 12 = 120$. Combine like terms: $6x + 12 = 120$. Subtract 12 from both sides: $6x = 120 - 12 = 108$. Divide both sides by 6: $x = \frac{108}{6} = 18$.

Step4: Find the rate of each ship

Slower ship: $x = 18$ knots. Faster ship: $x + 4 = 18 + 4 = 22$ knots.

Answer:

Machine A produces 3060 gadgets, Machine B produces 1530 gadgets, Machine C produces 1580 gadgets.

Problem 40