Question

A mass of \(176 \mathrm{~kg}\) is divided in the ratio 4:5:7. Calculate the mass of each portion.

Short answer

Answer: The masses of the 3 portions are 44 kg, 55 kg, and 77 kg.

Step-by-step solution

Calculate the sum of the ratios

To find the sum of the ratio 4:5:7, we simply add the numbers 4, 5, and 7 together: \(4 + 5 + 7 = 16\).

Determine the mass of each portion

Now that we know the sum of the ratios (16), we can calculate the mass of each portion. We will multiply the total mass (176 kg) by each portion's ratio and divide by the sum of the ratios (16). Portion 1 (ratio 4): \(\frac{176 \times 4}{16} = \frac{704}{16} = 44 \mathrm{~kg}\) Portion 2 (ratio 5): \(\frac{176 \times 5}{16} = \frac{880}{16} = 55 \mathrm{~kg}\) Portion 3 (ratio 7): \(\frac{176 \times 7}{16} = \frac{1232}{16} = 77 \mathrm{~kg}\) So, the masses of the 3 portions are 44 kg, 55 kg, and 77 kg.

Key concepts

Mass Division

Understanding the concept of mass division is important for distributing a total mass into smaller portions based on a given ratio. Mass division occurs when you take a whole, such as a weight or quantity, and split it according to specified proportions.
For example, if you have a mass of 176 kg that needs to be divided into ratios of 4:5:7, you must first identify how many parts or portions the total mass will be divided into.

• The first step is to add up the numbers in the ratio: 4 + 5 + 7 = 16 parts in total.
• These parts represent the shares into which you will divide the total mass.

Using the ratio parts, you can calculate the individual masses by distributing the mass proportionally. Hence, mass division requires adding the ratio numbers to find the total parts and understanding that these parts dictate how the mass is shared.

Proportional Reasoning

Proportional reasoning involves using the relationship between numbers to solve problems such as dividing or scaling. It's especially useful in ratio problems, where you compare portions of a whole.
To apply proportional reasoning in our example, once you've calculated the total number of parts (16), you then determine what fraction of the mass each part represents.

• Each part of the ratio corresponds to an equal division of the total mass. In our example, 176 kg is divided by 16 parts.
• You then calculate the mass each portion holds by multiplying the total mass by each ratio value and dividing by the total number of parts (16).

Understanding proportional reasoning is crucial because it extends beyond simple addition or subtraction. It involves relationships between variables and helps in making sense of how changes in one part affect the whole.

Step-by-Step Problem Solving

Approaching problems through step-by-step problem solving ensures clarity and accuracy. This methodical approach is particularly beneficial in mathematical calculations involving ratios.
For our mass division problem:

• First, sum the ratios to understand the whole picture. The numbers in the ratio are your starting point.
• Second, use this total to determine the value of each ratio portion by multiplying the total mass (176 kg) by each ratio number and dividing by the sum of the ratios (16).
• Finally, verify your results by ensuring the lower calculated masses add back to the total mass of 176 kg.

This helps in fostering a logical sequence, minimizing errors, and ensuring that you do not miss any critical steps in your calculations. By honing these skills, you'll become adept at tackling more complex problems with confidence.