Chapter 2: Introduction to Linear Polynomials

Exercise

Question

Question: 6

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The digits of a two-digit number differ by 3. If the digits are interchanged, and the resulting number is added to the original number, we get 143. Find both the numbers.

Answer

1
Let the tens digit be t and the units digit be u. Since the digits differ by 3, form an equation.
\text{$t - u = 3$ $\quad$ ${color:#6D28D9}[Equation 1]{/color}$ $\quad$ ${color:#625B5F}[Hint: Assume the tens digit is greater than the units digit.]{/color}$}
2
Express the original number and the number formed by interchanging its digits in terms of t and u.
Original number: $10t + u$
\text{Interchanged number: $10u + t$ $\quad$ ${color:#625B5F}[The units digit becomes the tens digit, and the tens digit becomes the units digit.]{/color}$}
3
Form an equation using the sum of the original number and the number formed by interchanging its digits.
$(10t + u) + (10u + t) = 143$
⇒
$11t + 11u = 143$
⇒
\text{ $t + u = 13$ $\quad$ ${color:#6D28D9}[Equation 2]{/color}$ $\quad$ ${color:#625B5F}[Divide both sides by 11]{/color}$}
4
Add Equations 1 and 2 to eliminate u and find t.
$(t - u) + (t + u) = 3 + 13$
⇒
$2t = 16$
⇒
$t = 8$
5
Substitute the value of t into Equation 1 to find u.
\text{${color:#2E2EFF}8{/color} - u = 3$}
⇒
$8 - 3 = u$
⇒
$u = 5$
6
Substitute the values of t and u to find the original and interchanged numbers.
Since $t = 8$ and $u = 5$:
Original number = $10t + u = 10 × 8 + 5 = 85$
Interchanged number = $10u + t = 10 × 5 + 8 = 58$
7
Write both numbers.
Answer: The two numbers are 85 and 58.
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