Math Problem Statement

Miko likes arranging fruit platters. In a fruit platter competition Miko joined, the ratio of apples to papayas the organiser prepared was 1: 5. If 92 papayas were added, the ratio of apples to papayas would be 1: 7. How many apples were there?

Solution

Let the number of apples be aa, and the number of papayas be pp.

We are given the following two pieces of information:

  1. Initially, the ratio of apples to papayas is 1:5, which means:

ap=15ora=15p\frac{a}{p} = \frac{1}{5} \quad \text{or} \quad a = \frac{1}{5}p

  1. After 92 papayas are added, the new ratio of apples to papayas becomes 1:7, so:

ap+92=17ora=17(p+92)\frac{a}{p + 92} = \frac{1}{7} \quad \text{or} \quad a = \frac{1}{7}(p + 92)

Step 1: Express aa in terms of pp

From the first equation, we know that a=15pa = \frac{1}{5}p.

From the second equation, we have a=17(p+92)a = \frac{1}{7}(p + 92).

Since both expressions represent aa, we can set them equal to each other:

15p=17(p+92)\frac{1}{5}p = \frac{1}{7}(p + 92)

Step 2: Solve for pp

Multiply both sides by 35 to eliminate the denominators:

7p=5(p+92)7p = 5(p + 92)

Expand the right side:

7p=5p+4607p = 5p + 460

Subtract 5p5p from both sides:

2p=4602p = 460

Solve for pp:

p=230p = 230

Step 3: Find aa

Now that we know p=230p = 230, substitute this value into the first equation to find aa:

a=15×230=46a = \frac{1}{5} \times 230 = 46

Thus, the number of apples is 46\boxed{46}.

Would you like further clarification or details on the solution?

Here are 5 related questions to explore:

  1. If 10 more papayas were added to the final scenario, what would the new ratio be?
  2. How many papayas were there initially in the competition?
  3. What if the initial ratio was 1:4 instead of 1:5, and the same number of papayas were added?
  4. What is the percentage increase in the number of papayas after 92 were added?
  5. How would the ratios change if 20 apples were added instead?

Tip: When solving ratio problems, it's helpful to express unknowns in terms of one variable and use systems of equations to simplify the solution.

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Math Problem Analysis

Mathematical Concepts

Ratios
Algebra
Systems of Equations

Formulas

Ratio formula: a/p = 1/5
a = 1/5 * p
a = 1/7 * (p + 92)
Solving linear equations

Theorems

Basic ratio properties
Equation solving methods

Suitable Grade Level

Grades 6-8