Question

A two digit number is four times of the sum of its digits and twice the product of its digits. Find the number.

Hint:

Always remember to represent a two-digit number with digits 't' and 'u' as \(10t + u\). A common mistake is to write it as \(tu\), which implies multiplication.

Answer 1

Step 1: Understanding the Concept:
We need to translate the given word problem into a system of two algebraic equations with two variables, representing the tens and units digits of the number.

Step 2: Setting up the Equations:
Let the tens digit be \(t\) and the units digit be \(u\).
The value of the two-digit number can be represented as \(10t + u\).
The sum of the digits is \(t + u\).
The product of the digits is \(t \cdot u\).
From the problem statement, we get two conditions:
Condition 1: The number is four times the sum of its digits. \[ 10t + u = 4(t + u) \] Condition 2: The number is twice the product of its digits. \[ 10t + u = 2(t \cdot u) \]

Step 3: Solving the System of Equations:
Let's first simplify Equation 1: \[ 10t + u = 4t + 4u \] \[ 6t = 3u \] \[ u = 2t \] Now, substitute this expression for \(u\) into Equation 2: \[ 10t + (2t) = 2(t \cdot (2t)) \] \[ 12t = 2(2t^2) \] \[ 12t = 4t^2 \] Rearrange into a quadratic form: \[ 4t^2 - 12t = 0 \] \[ 4t(t - 3) = 0 \] This gives two possible solutions for \(t\): \(t=0\) or \(t=3\).
Since it is a two-digit number, the tens digit \(t\) cannot be 0. So, we must have \(t = 3\).
Now, find the units digit \(u\): \[ u = 2t = 2(3) = 6 \] The digits are \(t=3\) and \(u=6\). The number is \(10(3) + 6 = 36\).
Step 4: Final Answer and Verification:
The number is 36.
Check: Sum of digits = 3 + 6 = 9. Four times the sum = \(4 \times 9 = 36\). (Condition 1 is met) Product of digits = 3 \(\times\) 6 = 18. Twice the product = \(2 \times 18 = 36\). (Condition 2 is met) The answer is correct.