Solving Ratio Problems: Finding the Number of Pencils in a Box

Have you ever encountered a problem that asks you to find the number of pencils in a box based on the ratio of pencils to pens? This type of problem is a classic example of a ratio problem. Let's dive into a specific example to understand how to solve it step by step.

Understanding the Problem Statement

The problem states that there is a box containing pencils and pens in the ratio of 5 pencils for every 7 pens. It also mentions that there are 42 pens in the box. The question asks how many pencils are in the box.

Setting Up the Ratio

First, we need to understand the ratio given:

5 pencils for every 7 pens

This can be written as a ratio: 5:7.

Using this ratio, we can establish a relationship between the number of pencils and pens in the box. Let's denote the number of pencils as P and the number of pens as Q.

The given ratio can be written as:

[frac{P}{Q} frac{5}{7}]

Given that Q (the number of pens) is 42, we can substitute Q with 42 in the ratio:

[frac{P}{42} frac{5}{7}]

Calculating the Number of Groups of Pens

To solve for P (the number of pencils), we need to determine the number of groups of pens in the box. We can do this by dividing the total number of pens by the number of pens in each group.

[text{Number of groups} frac{text{Number of pens}}{text{Pens per group}} frac{42}{7} 6]

This means there are 6 groups of 7 pens each in the box.

Calculating the Number of Pencils

Now, we can use the fact that there are 5 pencils for each group of 7 pens to find the total number of pencils. We multiply the number of groups by the number of pencils per group:

[text{Number of pencils} text{Number of groups} times text{Pencils per group} 6 times 5 30]

Therefore, there are 30 pencils in the box.

Verifying the Solution

To confirm our solution, we can use the following verification method:

[frac{30}{42} frac{5}{7}]

This shows that the ratio of 30 pencils to 42 pens is indeed 5:7, confirming our solution is correct.

Conclusion

Ratio problems like this can be solved by understanding the ratio given, breaking down the problem into smaller steps, and methodically working through the calculations. Understanding ratios and proportions is essential in many real-world applications, from cooking to construction.

By practicing similar problems, you can become more comfortable with ratios and improve your problem-solving skills. Keep practicing and stay curious!