Question

The sum of two numbers is 16. Thrice the smaller number is greater than twice the larger by 3. The product of the numbers is ?

Hint:

When solving word problems, clearly define your variables (e.g., let x be the smaller number). This helps in accurately translating the sentences into mathematical equations and avoiding confusion.

Answer 1

Step 1: Understanding the Concept:
This word problem can be solved by translating the given statements into a system of two linear equations with two variables and then solving for the variables.
Step 2: Detailed Explanation:
Let the two numbers be \(x\) and \(y\). Let \(x\) be the smaller number and \(y\) be the larger number.
From the first statement: "The sum of two numbers is 16." \[ x + y = 16 \quad \text{(Equation 1)} \] From the second statement: "Thrice the smaller number is greater than twice the larger by 3." \[ 3x = 2y + 3 \quad \text{(Equation 2)} \] Now, we solve this system of equations. From Equation 1, we can express \(y\) in terms of \(x\): \[ y = 16 - x \] Substitute this expression for \(y\) into Equation 2: \[ 3x = 2(16 - x) + 3 \] \[ 3x = 32 - 2x + 3 \] \[ 3x = 35 - 2x \] Add \(2x\) to both sides: \[ 5x = 35 \] \[ x = 7 \] Now that we have the value of the smaller number, \(x\), we can find the larger number, \(y\), using Equation 1: \[ y = 16 - x = 16 - 7 = 9 \] The two numbers are 7 and 9.
Step 3: Final Answer:
The question asks for the product of the numbers. \[ \text{Product} = x \times y = 7 \times 9 = 63 \]