An artisan chocolate factory produces a signature dark chocolate blend using cocoa mass, c… — GMAT

An artisan chocolate factory produces a signature dark chocolate blend using cocoa mass, cocoa butter, and cane sugar in the initial ratio of $2 : 3 : 4$ by weight, respectively. To modify the flavor profile for a new batch, the master chocolatier adds $10$ kilograms of cocoa mass and $20$ kilograms of cocoa butter to the mix, while removing $5$ kilograms of cane sugar. As a result, the ratio of cocoa mass to cocoa butter in the modified mixture becomes $4 : 7$ . What was the total weight, in kilograms, of the initial dark chocolate blend?

A
$25$
B
$35$
$45$ Answer
D
$60$
E
$70$

Answer

The total weight of the initial dark chocolate blend was

45

kilograms.

By setting the initial weights of cocoa mass, cocoa butter, and cane sugar as

2x

3x

, and

4x

, the initial total weight is

9x

. The modified weights for cocoa mass and cocoa butter are

2x + 10

and

3x + 20

, respectively. Equating their ratio to

\frac{4}{7}

gives

\frac{2x + 10}{3x + 20} = \frac{4}{7}

, which simplifies to

14x + 70 = 12x + 80

, so

2x = 10

and

x = 5

. Substituting

x = 5

into the total weight expression

9x

yields

9(5) = 45

kilograms.

Step-by-Step Solution

Define initial ingredient weights using a multiplier variable.

Cocoa mass =

2x

, Cocoa butter =

3x

, Cane sugar =

4x

. Total initial weight =

2x + 3x + 4x = 9x

Expressing quantities in terms of a common ratio multiplier

x

enables setting up algebraic equations after alterations.

Express the modified weights of cocoa mass and cocoa butter.

New cocoa mass =

2x + 10

; New cocoa butter =

3x + 20

The problem states

10

kg of cocoa mass and

20

kg of cocoa butter were added.

Set up the proportion equation using the new ratio of cocoa mass to cocoa butter.

\frac{2x + 10}{3x + 20} = \frac{4}{7}

The modified ratio of cocoa mass to cocoa butter is given as

4 : 7

Cross-multiply and solve for

x

7(2x + 10) = 4(3x + 20) \implies 14x + 70 = 12x + 80 \implies 2x = 10 \implies x = 5

Solving the linear equation determines the common ratio multiplier value.

Calculate the initial total weight of the blend.

Initial total weight =

9x = 9(5) = 45

kg.

Multiplying the sum of all initial ratio parts by the multiplier gives the requested initial total weight.

Key Concept

Ratio setup and algebraic scaling in multi-part word problems with changing quantities

Estimated Time:2m 0s

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