Question
Suppose that on the shelf in a supermarket, the ratio of the number of small milk cartons ( 1 litre) to large milk cartons ( 2 litres) are in the ratio \( 5: 7 \). If there is a total of 180 cartons on the shelf, how many of each type are there?
Ask by Mathis Ray. in South Africa
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
There are 75 small cartons and 105 large cartons on the shelf.
Solution
Let's solve the problem step by step.
**Given:**
- The ratio of small milk cartons (1 litre) to large milk cartons (2 litres) is \( 5:7 \).
- The total number of cartons is 180.
**Let:**
- \( s \) = number of small cartons
- \( l \) = number of large cartons
**From the ratio \( 5:7 \):**
\[
\frac{s}{l} = \frac{5}{7}
\]
\[
s = \frac{5}{7}l
\]
**Total cartons:**
\[
s + l = 180
\]
**Substitute \( s \) from the ratio into the total:**
\[
\frac{5}{7}l + l = 180
\]
\[
\left(\frac{5}{7} + 1\right) l = 180
\]
\[
\frac{12}{7}l = 180
\]
\[
l = 180 \times \frac{7}{12} = 105
\]
**Find \( s \):**
\[
s = 180 - l = 180 - 105 = 75
\]
**Answer:**
- **Small cartons (1 litre):** 75
- **Large cartons (2 litres):** 105
Reviewed and approved by the UpStudy tutoring team
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Explain 
Simplify this solution Beyond the Answer
To solve this problem, let’s begin by defining the number of small milk cartons as \( 5x \) and the number of large milk cartons as \( 7x \), based on the ratio of \( 5:7 \). The total number of cartons is given as 180, so we can set up the equation: \[ 5x + 7x = 180 \] Combining the terms gives: \[ 12x = 180 \] Now, solving for \( x \) we get: \[ x = 15 \] Now we can find the number of each type of milk carton. For the small cartons: \[ 5x = 5 \times 15 = 75 \] And for the large cartons: \[ 7x = 7 \times 15 = 105 \] So, there are 75 small milk cartons and 105 large milk cartons on the shelf!
