A certain two-digit number is equivalent to five times the sum of the digits. It is found to be 9 less than…

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A certain two-digit number is equivalent to five times the sum of the digits. It is found to be 9 less than the number formed when the digits are interchanged. Find the number.

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13. A certain two-digit number is equivalent to five times the sum of the digits. It is found to be 9 less than the number formed when the digits are interchanged. Find the number.

Define the two-digit number using variables.
Let the two-digit number be $10t + u$ , where $t$ is the tens digit and $u$ is the units digit.
Formulate equations based on the given conditions.
Condition 1: The number is equivalent to five times the sum of its digits.
$10t + u = 5(t + u)$ $10t + u = 5t + 5u$ $10t - 5t = 5u - u$ $5t = 4u \quad \text{(Equation 1)}$
Condition 2: The number is 9 less than the number formed when the digits are interchanged.
The number with digits interchanged is $10u + t$ .
$10t + u = (10u + t) - 9$ $10t - t = 10u - u - 9$ $9t = 9u - 9$
Divide by 9:
$t = u - 1 \quad \text{(Equation 2)}$
Solve the system of equations.
Substitute Equation 2 into Equation 1:
$5(u - 1) = 4u$ $5u - 5 = 4u$ $5u - 4u = 5$ $u = 5$
Substitute the value of $u$ back into Equation 2 to find $t$ :
$t = 5 - 1$ $t = 4$
Form the number.
The tens digit is 4 and the units digit is 5.
The number is $10(4) + 5 = 40 + 5 = 45$ .
The number is 45.

A certain two-digit number is equivalent to five times the sum of the digits. It is found to be 9 less than the number formed when the digits are interchanged. Find the number.

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Step-by-step solution

Define the two-digit number using variables.

Formulate equations based on the given conditions.

Solve the system of equations.

Form the number.

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