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Mindy and Troy combined ate 9 pieces of wedding cake. Mindy ate 3 pieces of cake and Troy had (1)/(4) of the total cake. How many pieces of cake were there in all? □

Mindy and Troy combined ate 99 pieces of wedding cake. Mindy ate 33 pieces of cake and Troy had 14\frac{1}{4} of the total cake. How many pieces of cake were there in all? \newline□\square

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Full solution

Q. Mindy and Troy combined ate 99 pieces of wedding cake. Mindy ate 33 pieces of cake and Troy had 14\frac{1}{4} of the total cake. How many pieces of cake were there in all? \newline□\square
  1. Calculate Mindy and Troy's total: Determine the total number of pieces Mindy and Troy ate together.\newlineMindy ate 33 pieces of cake. According to the problem, Mindy and Troy together ate 99 pieces of cake.\newlineCalculation: 99 pieces (total eaten by both) −- 33 pieces (eaten by Mindy) == 66 pieces (eaten by Troy).
  2. Express Troy's share as fraction: Express Troy's share of the cake as a fraction of the total number of pieces. Troy had 14\frac{1}{4} of the total cake. If we let TT be the total number of pieces of cake, then Troy's share is (14)T\left(\frac{1}{4}\right)T.
  3. Set up equation for total: Set up an equation to find the total number of pieces of cake.\newlineFrom Step 11, we know Troy ate 66 pieces. From Step 22, we know that 66 pieces represent 14\frac{1}{4} of the total cake.\newlineSo, we can write the equation: 14T=6\frac{1}{4}T = 6.
  4. Solve for total number: Solve the equation for TT to find the total number of pieces of cake.\newlineMultiply both sides of the equation by 44 to solve for TT.\newline4×(14)T=4×64 \times (\frac{1}{4})T = 4 \times 6\newlineT=24T = 24
  5. Verify solution: Verify that the solution makes sense in the context of the problem.\newlineIf the total number of pieces of cake is 2424, then Troy's share would indeed be 14\frac{1}{4} of 2424, which is 66 pieces. Adding Mindy's 33 pieces to Troy's 66 pieces gives us 99 pieces, which matches the information given in the problem.

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