Understanding the Weight Problem

This question asks us to find the difference between the weights of two individuals, A and B, given two pieces of information about their weights. We are told their combined weight and a relationship between a percentage of A's weight and a fraction of B's weight.

Setting Up the Equations

Let A represent the weight of person A in kilograms and B represent the weight of person B in kilograms. We can translate the given information into two mathematical equations:

• Equation 1: The sum of weights of A and B is 80 kg.

  \(A + B = 80\)

• Equation 2: 50% of A's weight is \(\frac 5 6\) times the weight of B.

  50% of A can be written as \(\frac{50}{100}A\) or \(\frac{1}{2}A\).

  So, the equation is \(\frac{1}{2}A = \frac{5}{6}B\).

Solving the System of Equations

We now have a system of two linear equations with two variables:

1. \(A + B = 80\)
2. \(\frac{1}{2}A = \frac{5}{6}B\)

We can solve this system using the substitution method. Let's simplify Equation 2 first:

Multiply both sides of Equation 2 by 2 to isolate A:

\(2 \times \frac{1}{2}A = 2 \times \frac{5}{6}B\)

\(A = \frac{10}{6}B\)

\(A = \frac{5}{3}B\)

Now substitute this expression for A into Equation 1:

\(\frac{5}{3}B + B = 80\)

To add \(\frac{5}{3}B\) and B, we need a common denominator. B is the same as \(\frac{3}{3}B\):

\(\frac{5}{3}B + \frac{3}{3}B = 80\)

\(\frac{5+3}{3}B = 80\)

\(\frac{8}{3}B = 80\)

Now, solve for B by multiplying both sides by \(\frac{3}{8}\):

\(B = 80 \times \frac{3}{8}\)

\(B = \frac{80}{8} \times 3\)

\(B = 10 \times 3\)

\(B = 30\)

So, the weight of B is 30 kg.

Now substitute the value of B (30) back into Equation 1 (\(A + B = 80\)) to find A:

\(A + 30 = 80\)

\(A = 80 - 30\)

\(A = 50\)

So, the weight of A is 50 kg.

Finding the Difference in Weights

The question asks for the difference between their weights. This is the absolute difference between A and B.

Difference = \(|A - B|\)

Difference = \(|50 - 30|\)

Difference = \(|20|\)

Difference = 20 kg

Verification

Let's check if our calculated weights satisfy the original conditions:

• Is the sum of weights 80 kg? \(50 + 30 = 80\). Yes.
• Is 50% of A equal to \(\frac{5}{6}\) times B?

  50% of A = 0.50 * 50 kg = 25 kg.

  \(\frac{5}{6}\) times B = \(\frac{5}{6} \times 30\) kg = \(5 \times 5\) kg = 25 kg.

  Yes, 25 kg = 25 kg.

Both conditions are met, so our calculated weights and their difference are correct.

Conclusion

The difference between the weights of A and B is 20 kg.

Revision Table: Key Concepts

Additional Information: Solving Word Problems

Word problems often require translating the given information into mathematical equations. Here are some tips:

• Read the problem carefully to understand what is given and what needs to be found.
• Assign variables to the unknown quantities.
• Write down the equations based on the relationships described in the problem.
• Choose an appropriate method to solve the system of equations (e.g., substitution, elimination).
• Solve the equations to find the values of the variables.
• Check your answer by plugging the values back into the original word problem or equations.
• Make sure your final answer is in the correct units and answers the specific question asked.