Solving Mathematical Problems: A Practical Approach to Ratio and Proportion

Mathematics is a subject that often involves solving problems using various techniques and principles. One such common type of problem involves ratios, where we need to find the quantities in a given ratio. Today, we will explore a practical example of solving a ratio problem using a simple and systematic approach. We will use the example of Ed and his 30 sweets to illustrate the steps involved.

Understanding the Problem

The given problem is: Ed has 30 sweets, and the ratio of red sweets to yellow sweets is 2:3. Our goal is to find out how many red sweets Ed has.

Step-by-Step Solution

Let's break down the problem into manageable steps:

Step 1: Identify the Total Parts in the Ratio

First, we need to identify the total parts in the given ratio. The ratio of red sweets to yellow sweets is 2:3. To find the total parts, we simply add the two numbers in the ratio:

Total parts 2 3 5

Step 2: Determine the Value of Each Part

Next, we need to find the value of each part by dividing the total number of sweets by the total number of parts:

Value of each part Total sweets / Total parts

Value of each part 30 / 5 6

Step 3: Calculate the Number of Red Sweets

Finally, we can calculate the number of red sweets by multiplying the number of parts for red sweets by the value of each part:

Number of red sweets 2 parts × 6 12

Therefore, Ed has 12 red sweets.

Verifying the Solution

To ensure our solution is correct, we can verify by following another method:

Alternate Method

1. Divide the total sweets into the total parts of the ratio. The total parts are 5.

2. Calculate the value of one part:

3. Value of one part Total sweets / Total parts

3. Value of one part 30 / 5 6

4. Calculate the number of red sweets, which is 2 parts of the total:

5. Number of red sweets 2 × 6 12

The result is the same, confirming our solution is correct.

Additional Practice

To further solidify your understanding, let's consider an additional example:

If the ratio of two quantities is 3:5 and the total value of these quantities is 48, we can solve for the individual values as follows:

Solution:

Total parts in the ratio 3 5 8

Value of each part Total value / Total parts 48 / 8 6

For the first quantity (3 parts):

Value 3 × 6 18

For the second quantity (5 parts):

Value 5 × 6 30

Thus, the first quantity is 18 and the second quantity is 30.

Conclusion

Understanding and applying ratios is a fundamental skill in mathematics. By following a systematic approach, we can easily solve problems like the one with Ed and his sweets. Practice similar problems to build your confidence and proficiency in solving ratio and proportion problems.