A two-digit number is 5 times the sum of its digits. Adding 9 to the number reverses its digits. Find the number.

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations

Step-by-Step Solution

Let the tens digit be

x

and the units digit

y

, so the number is

10 x + y

. First condition:

10 x + y = 5 (x + y)

→

5 x = 4 y

. Second:

10 x + y + 9 = 10 y + x

→

9 x + 9 = 9 y

→

y = x + 1

. Then

5 x = 4 (x + 1)

→

x = 4, y = 5

, giving 45. Check: 4 + 5 = 9 and

5 \times 9 = 45

✓; 45 + 9 = 54, the reverse ✓. Picking 54 reports the REVERSED number instead of the original.

Answer: D) 45

Key idea: Word problems and translation

Most Class 10 questions on this topic are word problems. The skill is in TRANSLATION:

1. Identify the two unknowns; assign variables (e.g., let

x

= age of A,

y

= age of B).
2. Write two equations from the two given pieces of information.
3. Solve by your preferred method.

**Common types**:
- **Age problems**: 'Father is twice as old as son. Five years ago father was three times. Find ages.'
- **Number problems**: 'A two-digit number plus its reverse is 99...'
- **Distance/time/speed**: 'Boat in still water speeds, downstream/upstream rates...'
- **Cost problems**: 'Cost of pencils and pens.'
- **Fraction problems**: 'A fraction becomes 1/2 when 1 is added to numerator and 2 is subtracted from denominator...'

Always CHECK by substituting solution back into the original word problem (not just the equations).

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