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Difficulty: MediumRatio and Proportion Word Problems

An investment fund initially holds shares of Technology, Healthcare, and Energy stocks in the ratio 5:3:25 : 3 : 2, respectively. After the fund sells 3030 shares of Technology stock and purchases 3030 shares of Energy stock, the ratio of Technology, Healthcare, and Energy shares becomes 4:3:34 : 3 : 3. What was the total number of shares in the investment fund initially?

Answer: 300 shares

Answer

The initial total number of shares in the investment fund was 300.
The initial shares can be represented as 5x5x, 3x3x, and 2x2x for a total of 10x10x shares. After selling 3030 Technology shares and buying 3030 Energy shares, the Healthcare shares remain unchanged at 3x3x. Since Healthcare represents 33 parts in both the initial and final ratios (5:3:25:3:2 and 4:3:34:3:3), the value of one ratio part remains constant at xx. Therefore, the decrease in Technology shares from 5x5x to 4x4x corresponds directly to 1x=301x = 30 shares. Multiplying x=30x = 30 by the initial total of 1010 parts gives 300300 shares.

Step-by-Step Solution

1
Define initial quantities using a common ratio multiplier xx
Technology = 5x5x, Healthcare = 3x3x, Energy = 2x2x. Total initial shares = 10x10x.
Representing ratio terms algebraically preserves their proportional relationships.
2
Account for changes in share counts
New Technology shares = 5x−305x - 30, Healthcare shares = 3x3x, new Energy shares = 2x+302x + 30.
30 shares of Technology were sold and 30 shares of Energy were added.
3
Set up an equation comparing the new ratio of Technology to Healthcare
5x−303x=43\frac{5x - 30}{3x} = \frac{4}{3}
The new ratio of Technology to Healthcare is given as 4:34 : 3.
4
Solve for multiplier xx
3(5x−30)=12x  ⟹  15x−90=12x  ⟹  3x=90  ⟹  x=303(5x - 30) = 12x \implies 15x - 90 = 12x \implies 3x = 90 \implies x = 30
Cross-multiplying eliminates fractions to solve for the unknown multiplier.
5
Calculate the initial total number of shares
10x=10×30=30010x = 10 \times 30 = 300
The initial total is the sum of the initial parts (5x+3x+2x=10x5x + 3x + 2x = 10x).

Key Concept

Ratio Modification and Algebraic Scaling
Estimated Time:1m 30s
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Topics

Ratio and Proportion Word Problems

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