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Difficulty: MediumRatios, Rates, and Proportions

An event production company initially allocates its total equipment budget among Stage Lights, Ambient LED Strips, and Spotlight Towers in the ratio 4:5:34 : 5 : 3, respectively. Due to venue price adjustments, the budget allocated to Stage Lights is increased by 25%25\%, while the budget allocated to Spotlight Towers is reduced by 20%20\%. If the total overall budget remains unchanged, what is the new ratio of the budget allocated to Stage Lights to Ambient LED Strips to Spotlight Towers?

  1. 25:23:1225 : 23 : 12Answer
  2. B
    25:25:1225 : 25 : 12
  3. C
    25:17:1225 : 17 : 12
  4. D
    35:22:335 : 22 : 3
  5. E
    5:4:35 : 4 : 3

Answer

The new ratio of the budget allocated to Stage Lights to Ambient LED Strips to Spotlight Towers is 25:23:1225 : 23 : 12.
By setting the initial allocations to 4x4x, 5x5x, and 3x3x, the total budget is 12x12x. The new allocation for Stage Lights becomes 4x×1.25=5x4x \times 1.25 = 5x, and for Spotlight Towers it becomes 3x×0.80=2.4x3x \times 0.80 = 2.4x. To keep the total budget at 12x12x, the Ambient LED Strips allocation must be 12x−5x−2.4x=4.6x12x - 5x - 2.4x = 4.6x. The ratio 5:4.6:2.45 : 4.6 : 2.4 scales up to 50:46:2450 : 46 : 24, which simplifies to 25:23:1225 : 23 : 12.

Step-by-Step Solution

1
Represent the initial component allocations and the total budget in terms of a common variable.
Let the initial allocations be Stage Lights =4x= 4x, Ambient LED Strips =5x= 5x, and Spotlight Towers =3x= 3x. Total budget =4x+5x+3x=12x= 4x + 5x + 3x = 12x.
Establishing algebraic quantities allows precise tracking of individual changes relative to the constant total.
2
Calculate the updated allocations for Stage Lights and Spotlight Towers after their respective percentage changes.
New Stage Lights =4x×(1+0.25)=5x= 4x \times (1 + 0.25) = 5x.
New Spotlight Towers =3x×(1−0.20)=2.4x= 3x \times (1 - 0.20) = 2.4x.
Applying the specified percentage shifts directly to their corresponding original component values.
3
Determine the new allocation for Ambient LED Strips using the fixed total budget condition.
New Ambient LED Strips =12x−(5x+2.4x)=12x−7.4x=4.6x= 12x - (5x + 2.4x) = 12x - 7.4x = 4.6x.
Because the total overall budget remains unchanged at 12x12x, the sum of all three new allocations must equal 12x12x.
4
Write the new ratio and simplify to integer terms.
Ratio =5x:4.6x:2.4x=5:4.6:2.4=50:46:24=25:23:12= 5x : 4.6x : 2.4x = 5 : 4.6 : 2.4 = 50 : 46 : 24 = 25 : 23 : 12.
Multiplying by 1010 clears decimals, and dividing by 22 reduces the ratio to its simplest integer form.

Key Concept

Multi-part ratio adjustment with percentage changes under a constant total constraint
Estimated Time:1m 30s
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Topics

Ratios, Rates, and Proportions

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