🔍 Solving Money Mysteries: Unraveling Coin & Wage Puzzles 🧩💰

Solution for Number of Plain and Fancy Pennants

Let P be the number of plain pennants and F be the number of fancy pennants.

Equation 1 (Based on the total number of students): P + F = 30

Equation 2 (Based on the total cost): 4P + 8F = 168

Using the Elimination Method:

  1. Multiply the first equation by 4 to make it 4P + 4F = 120
  2. Subtract this new equation from the second equation: (4P + 8F) – (4P + 4F) = 168 – 120, resulting in 4F = 48
  3. Divide by 4 to find F: F = 12
  4. Substitute F = 12 into P + F = 30 to find P: P = 18

So 18 plain pennants and 12 fancy pennants were bought.

Solution for Number of Adult and Student Tickets

Let A be the number of adult tickets and S be the number of student tickets.

Equation 1 (Based on the total number of tickets): A + S = 920

Equation 2 (Based on the total revenue): 4A + 2S = 2446

Using the Elimination Method:

  1. Multiply the first equation by 2 to make it 2A + 2S = 1840
  2. Subtract this new equation from the second equation: (4A + 2S) – (2A + 2S) = 2446 – 1840, resulting in 2A = 606
  3. Divide by 2 to find A: A = 303
  4. Substitute A = 303 into A + S = 920 to find S: S = 617

So 303 adult tickets and 617 student tickets were sold.

Solution for Number of Dimes and Nickels

Let D be the number of dimes and N be the number of nickels.

Equation 1 (Based on the total number of coins): D + N = 40

Equation 2 (Based on the total value in cents): 10D + 5N = 290

Using the Elimination Method:

  1. Multiply the first equation by 5 to make it 5D + 5N = 200.
  2. Subtract this new equation from the second equation: (10D + 5N) – (5D + 5N) = 290 – 200, resulting in 5D = 90.
  3. Divide by 5 to find D: D = 18.
  4. Substitute D = 18 into D + N = 40 to find N: N = 22.

So 18 dimes and 22 nickels were used.

Solution for Celia’s Apple Business

Let C be the cost of one apple in cents.

Let S be the selling price of one apple in cents.

Step 1: Celia initially bought 12 apples, so the total cost is 12C.

Step 2: Celia ate 2 apples, leaving her with 10 apples.

Step 3: She sold each of these 10 apples for S cents, making the total revenue 10S.

Step 4: The selling price per apple is 20 cents more than the cost, so S = C + 20.

Step 5: Her profit is 1 dollar or 100 cents, which means 10S – 12C = 100.

Using the Elimination Method:

  1. Substitute the value of S from Step 4 into Step 5: 10(C + 20) – 12C = 100.
  2. Expand and simplify: 10C + 200 – 12C = 100.
  3. Combine like terms: -2C + 200 = 100.
  4. Solve for C: C = 50 cents.
  5. Use C to find S: S = 50 + 20 = 70 cents.

So the cost per apple is 50 cents, and she sold each for 70 cents.

Solution for Finding the Weekday Hourly Wage (W)

Given the total weekly pay T is 142 dollars, the equation becomes:

142 = 10W + 8W + 16

Steps to Solve:

  1. Combine like terms: 142 = 18W + 16
  2. Subtract 16 from both sides: 126 = 18W
  3. Divide by 18 to find W: W = 7

So the weekday hourly wage (W) is 7 dollars.

Solution for Number of Nickels, Dimes, and Quarters Warren Has

Equation 1 (Based on the total number of coins): N + D + Q = 40

Equation 2 (Based on the total value in cents): 5N + 10D + 25Q = 405

Equation 3 (Warren has 7 more nickels than dimes): N = D + 7

Using the Elimination and Substitution Methods:

  1. Substitute Equation 3 into Equation 1: (D + 7) + D + Q = 40, simplifying to 2D + Q = 33
  2. Substitute Equation 3 into Equation 2: 5(D + 7) + 10D + 25Q = 405, simplifying to 5D + 35 + 10D + 25Q = 405 or 15D + 25Q = 370
  3. Now solve the two new equations simultaneously: 2D + Q = 33 and 15D + 25Q = 370
  4. Multiply the first new equation by 25 and the second new equation by 1, making it easier to eliminate one variable.
  5. Subtract the second new equation from the first new one: 50D + 25Q – (15D + 25Q) = 825 – 370, simplifying to 35D = 455
  6. Solve for D: D = 13
  7. Substitute D = 13 into any of the new equations to find Q: Q = 33 – 2D = 33 – 26 = 7
  8. Substitute D = 13 into Equation 3 to find N: N = D + 7 = 13 + 7 = 20

So Warren has 20 nickels, 13 dimes, and 7 quarters.

Solution for Number of Nickels, Dimes, and Quarters Warren Has Using One Variable

Let D be the number of dimes.

Then N = D + 7 (number of nickels)

And Q = 40 – (D + N) (number of quarters)

Steps to Solve:

  1. Substitute N and Q into the total value equation: 5N + 10D + 25Q = 405 becomes 5(D + 7) + 10D + 25(40 – (D + D + 7)) = 405
  2. Simplify the equation: 5D + 35 + 10D + 1000 – 50D – 175 = 405
  3. Combine like terms: -35D + 860 = 405
  4. Solve for D: D = 13
  5. Find N using N = D + 7: N = 20
  6. Find Q using Q = 40 – (D + N): Q = 7

So Warren has 20 nickels, 13 dimes, and 7 quarters.