Math Monks worksheet with five word problems on systems of linear equations.

Here is the step-by-step solution to each of the 5 word problems from the worksheet.

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Problem 1: Boat Speed and Current

> A boat traveled 336 miles downstream and back. The trip downstream took 12 hours. The trip back took 14 hours. Find the speed of the boat in still water. What is the speed of the current?

Let:
- $ b $ = speed of the boat in still water (in mph)
- $ c $ = speed of the current (in mph)

Downstream: The boat moves *with* the current, so its effective speed is $ b + c $.
Distance = Speed × Time → $ 336 = (b + c) \cdot 12 $

Upstream: The boat moves *against* the current, so its effective speed is $ b - c $.
$ 336 = (b - c) \cdot 14 $

Set up the system:

Equation 1: $ 12(b + c) = 336 $
Equation 2: $ 14(b - c) = 336 $

Simplify both equations:

Divide Equation 1 by 12:
→ $ b + c = 28 $ ...(1)

Divide Equation 2 by 14:
→ $ b - c = 24 $ ...(2)

Add equations (1) and (2):

$ (b + c) + (b - c) = 28 + 24 $
→ $ 2b = 52 $
→ $ b = 26 $

Substitute into (1):

$ 26 + c = 28 $
→ $ c = 2 $

✔ Answer:
- Speed of the boat in still water: 26 mph
- Speed of the current: 2 mph

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Problem 2: Umbrellas and Shoes

> Michael bought 2 umbrellas and 3 pairs of shoes for $5.13. He then bought another umbrella and another 2 pairs of shoes for $3.09. Find the cost of each umbrella and each pair of shoes.

Let:
- $ u $ = cost of one umbrella (in dollars)
- $ s $ = cost of one pair of shoes (in dollars)

Set up the system:

Equation 1: $ 2u + 3s = 5.13 $
Equation 2: $ u + 2s = 3.09 $

Solve using substitution or elimination. Let’s use elimination.

Multiply Equation 2 by 2:
→ $ 2u + 4s = 6.18 $ ...(2a)

Now subtract Equation 1 from (2a):

$ (2u + 4s) - (2u + 3s) = 6.18 - 5.13 $
→ $ s = 1.05 $

Substitute into Equation 2:

$ u + 2(1.05) = 3.09 $
→ $ u + 2.10 = 3.09 $
→ $ u = 0.99 $

✔ Answer:
- Cost of each umbrella: $0.99
- Cost of each pair of shoes: $1.05

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Problem 3: Fraction Problem

> The denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are decreased by 6, then the denominator becomes 12 times the numerator. Determine the fraction.

Let:
- $ n $ = numerator
- $ d $ = denominator

From first sentence:
$ d = 2n + 4 $ ...(1)

From second sentence:
After decreasing both by 6: new numerator = $ n - 6 $, new denominator = $ d - 6 $, and:
$ d - 6 = 12(n - 6) $ ...(2)

Substitute (1) into (2):

$ (2n + 4) - 6 = 12(n - 6) $
→ $ 2n - 2 = 12n - 72 $

Solve for $ n $:

Bring variables to one side:

$ -2 + 72 = 12n - 2n $
→ $ 70 = 10n $
→ $ n = 7 $

Find $ d $:

$ d = 2(7) + 4 = 14 + 4 = 18 $

✔ Answer: The fraction is $ \boxed{\dfrac{7}{18}} $

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Problem 4: Two Numbers — Difference and Sum

> The difference of two numbers is 3. Their sum is 13. Find the numbers.

Let:
- $ x $ and $ y $ be the two numbers.

Equations:

$ x - y = 3 $ ...(1)
$ x + y = 13 $ ...(2)

Add equations (1) and (2):

$ (x - y) + (x + y) = 3 + 13 $
→ $ 2x = 16 $
→ $ x = 8 $

Substitute into (2):

$ 8 + y = 13 $
→ $ y = 5 $

✔ Answer: The numbers are $ \boxed{8} $ and $ \boxed{5} $

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Problem 5: Two Numbers — Sum and Difference

> Find the value of two numbers if their sum is 16 and their difference is 6.

Let:
- $ x $ and $ y $ be the two numbers.

Equations:

$ x + y = 16 $ ...(1)
$ x - y = 6 $ ...(2)

Add equations (1) and (2):

$ 2x = 22 $
→ $ x = 11 $

Substitute into (1):

$ 11 + y = 16 $
→ $ y = 5 $

✔ Answer: The numbers are $ \boxed{11} $ and $ \boxed{5} $

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## ✔ Final Answers Summary:

1. Boat & Current: Boat = 26 mph, Current = 2 mph
2. Umbrella & Shoes: Umbrella = $0.99, Shoes = $1.05
3. Fraction: $ \dfrac{7}{18} $
4. Numbers (diff 3, sum 13): 8 and 5
5. Numbers (sum 16, diff 6): 11 and 5

Let me know if you’d like to see graphs or alternative methods!