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Difficulty: MediumTranslating and Solving Algebraic Word Problems

A boutique chocolatier packages two types of gift boxes: Standard and Deluxe. Each Standard box contains 44 dark chocolates, and each Deluxe box contains 88 dark chocolates. The chocolatier plans to prepare a batch of boxes such that the number of Standard boxes is exactly twice the number of Deluxe boxes. If the chocolatier has a total of 160160 dark chocolates available and uses all of them for this batch, what is the total number of boxes (both Standard and Deluxe combined) they can make?

Answer: 30 boxes

Answer

The total number of boxes they can make is 30.
By defining the number of Deluxe boxes as xx, the number of Standard boxes is 2x2x. The equation for the total number of dark chocolates is 4(2x)+8x=1604(2x) + 8x = 160. Solving this gives 16x=16016x = 160, which simplifies to x=10x = 10. Thus, there are 1010 Deluxe boxes and 2020 Standard boxes, for a combined total of 3030 boxes.

Step-by-Step Solution

1
Define variables for the number of boxes of each type.
Let xx be the number of Deluxe boxes and 2x2x be the number of Standard boxes.
We are given that the number of Standard boxes is exactly twice the number of Deluxe boxes.
2
Set up an equation representing the total number of dark chocolates used.
4(2x)+8x=1604(2x) + 8x = 160
Each Standard box requires 44 dark chocolates, each Deluxe box requires 88, and the total used is 160160.
3
Solve the equation for xx.
16x=160⇒x=1016x = 160 \Rightarrow x = 10
Combining like terms gives 16x=16016x = 160, and dividing both sides by 1616 yields x=10x = 10.
4
Calculate the total number of boxes.
x+2x=30x + 2x = 30
The total number of boxes is the sum of Deluxe boxes (1010) and Standard boxes (2020).

Key Concept

Translating verbal descriptions of relationships and totals into linear equations to solve algebraic word problems.
Estimated Time:1m 30s
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Topics

Translating and Solving Algebraic Word Problems

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