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Difficulty: MediumRatio and Proportion Word Problems

A boutique perfumery blends three essential oils—Jasmine, Sandalwood, and Bergamot—in an initial volume ratio of 4:3:24 : 3 : 2, respectively. To modify the fragrance profile, the perfumer adds 20 milliliters20\text{ milliliters} of Sandalwood and 40 milliliters40\text{ milliliters} of Bergamot to the mixture, leaving the amount of Jasmine unchanged. If the resulting volume ratio of Sandalwood to Bergamot is 5:65 : 6, what was the total initial volume, in milliliters, of the fragrance batch?

Answer: 90 milliliters

Answer

The total initial volume of the fragrance batch was 90 milliliters.
By representing the initial amounts of Jasmine, Sandalwood, and Bergamot as 4x4x, 3x3x, and 2x2x respectively, the total initial volume is 9x9x. Setting up the proportion 3x+202x+40=56\frac{3x + 20}{2x + 40} = \frac{5}{6} yields x=10x = 10. Substituting x=10x = 10 into 9x9x gives the initial volume of 90 milliliters.

Step-by-Step Solution

1
Define variables for the initial quantities using the given ratio.
Jasmine volume = 4x4x, Sandalwood volume = 3x3x, Bergamot volume = 2x2x, where xx is a positive multiplier. Total initial volume = 4x+3x+2x=9x4x + 3x + 2x = 9x.
Representing ratio parts algebraically allows setting up equations for the modified amounts.
2
Formulate an equation based on the new ratio after adding essential oils.
New Sandalwood volume = 3x+203x + 20, New Bergamot volume = 2x+402x + 40. Equating their ratio to 5:65 : 6 gives 3x+202x+40=56\frac{3x + 20}{2x + 40} = \frac{5}{6}.
The problem specifies the new relationship between Sandalwood and Bergamot.
3
Solve the proportion for xx.
Cross-multiplying gives 6(3x+20)=5(2x+40)  ⟹  18x+120=10x+200  ⟹  8x=80  ⟹  x=106(3x + 20) = 5(2x + 40) \implies 18x + 120 = 10x + 200 \implies 8x = 80 \implies x = 10.
Finding the multiplier xx allows us to calculate the exact initial quantities.
4
Compute the total initial volume.
Total initial volume = 9x=9×10=909x = 9 \times 10 = 90 milliliters.
The question asks for the total initial volume of the batch.

Key Concept

Solving multi-step ratio problems involving additions to individual components by establishing an algebraic multiplier.
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Topics

Ratio and Proportion Word Problems

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