Question

The sum of two numbers is 50 and one number is \(\frac{7}{3}\) times of the other; then find the numbers.

Hint:

When translating word problems into equations, define your variables clearly. For example, "Let x be the larger number and y be the smaller number." This helps in correctly setting up relationships like \(x = \frac{7}{3}y\).

Answer 1

Step 1: Understanding the Concept:
This word problem can be solved by setting up a system of two linear equations with two variables based on the given information and then solving them simultaneously.

Step 2: Detailed Explanation:
Let the two numbers be \(x\) and \(y\).
From the problem statement, we can form two equations:
Equation 1: The sum of the two numbers is 50. \[ x + y = 50 \] Equation 2: One number is \(\frac{7}{3}\) times the other. \[ x = \frac{7}{3}y \] Now we can solve this system. We will use the substitution method by substituting the expression for \(x\) from Equation 2 into Equation 1.
\[ \left(\frac{7}{3}y\right) + y = 50 \] To solve for \(y\), find a common denominator: \[ \frac{7y}{3} + \frac{3y}{3} = 50 \] \[ \frac{10y}{3} = 50 \] Multiply both sides by 3: \[ 10y = 150 \] Divide by 10: \[ y = 15 \] Now substitute the value of \(y\) back into Equation 2 to find \(x\): \[ x = \frac{7}{3}(15) \] \[ x = 7 \times 5 \] \[ x = 35 \]

Step 3: Final Answer:
The two numbers are 35 and 15. We can check our answer: \(35 + 15 = 50\) and \(35 = \frac{7}{3} \times 15\). The conditions are met.