**Digit Word Problems **

- The tens digit of a certain number is 3 less than the units digit. The sum of the digits is 11. What is the number?

Solution

Let x = units digit

x – 3 = tens digit

**Equation**

x + (x – 3) = 11

2x – 3 = 11

2x = 14

**Answer**

x = 7 (units digit)

x – 3 = 4 (tens digit)

Using the values above for units and tens, we find the number 47.

- The tens digit of a number is twice the units digit. If the digits are reversed, the new number is 27 less than the original. Find the original number.

Solution

Let x = units digit

2x = tens digits

Then the original number is 10(2x) + x, the reserved number is 10(x) + 2x, and the new number is the original number less than 27.

**Equation**

10(x) + 2x = 10(2x) + x – 27

12x = 21x – 27

-9x = -27

x = 3 (units digit)

2x = 6 (tens digit)

**Answer**

The number is (6 X 10) + 3 or 63.

- The sum of the digits in a two-digit number is 12. If the digits are reversed, the number is 18 greater than the original number. What is the number?

Solution

Let x = units digit

12 – x = tens digit

Then the original number is 10(12 – x) + x and the reserved number is 10(x) + (12 – x).

**Equation:** The reserved number is the original number plus 18.

10(x) + (12 – x) = 10(12 – x) + (x) + 18

10x + 12 – x = 120 – 10x + x + 18

9x + 12 = 138 – 9x

18x = 126

x = 7 (units digit)

12 – x = 5 (tens digit)

**Answer**

The number is ( 5 x 10) + 7 or 57

**Check:**

10(7) + (12 – 7) = 10(12- 7) + (7) + 18

70 + 5 = 50 + 7 + 18

75 = 75

- The tens digit of a certain number is 5 more than the units digit. The sum of the digits is 9. Find the number.

Solution

Let x = units digit

x + 5 = tens digit

**Equation**

x + (x + 5) = 9

2x + 5 = 9

2x = 4

x = 2 (unit digit)

x + 5 = 7 (tens digit)

**Answer**

The number is (7 X 10) + 2 or 72.

- The tens digit of a two-digit number is twice the units digit. If the digits are reversed, the new number is 36 less than the original number. Find the number.

Solution

Let x = units digit

2x = tens digit

Then the number is 10(2x) + x and the reversed number is 10(x)+ 2x.

**Equation**

10(x) + 2x= 10(2x) + x – 36

12x = 21x – 36

-9x = -36

x = 4 (units digit)

2x = 8 (tens digit)

**Answer**

The number is (8 X 10) + 4 or 84.

- The sum of the digits of a two-digit number is 13. The units digit is 1 more than twice the tens digit. Find the number.

Solution

Let x = units digit

13 – x = tens digit

**Equation**

The units digit is twice the tens digit plus 1.

x = 2(13 – x) + 1

x = 26 – 2x + 1

x = 27 – 2x

3x = 27

x = 9 (units digit)

13 – x = 4 (tens digit)

**Answer**

The number is (4 x 10) + 9 or 49.

- The sum of the digits of a three-digit number is 6. The hundreds digit is twice the units digit, and the tens digit equals the sum of the other two. Find the number.

Solution

Let x = units digit

2x = hundreds digit

x + 2x = tens digit

**Equation**

x + 2x + (x + 2x) = 6

6x = 6

x = 1 (units digit)

2x = 2 (hundreds digit)

x + 2x = 3 (tens digit)

**So,**

The number is (2 x 100) + (3 x 10) + 1 or 231.

- The units digit is twice the tens digit. If the number is doubled, it will be 12 more than the reversed number. Find the number.

Solution

Let x = tens digit.

2x = units digit.

Then the number is 10(x) + 2x and the reversed number is 10(2) = x.

**Equation**

Two times the number equals 12 more than the reversed number.

2[10(x) + (2x)] = 10(2x) + x + 12

2(12x) = 21x + 12

24x = 21x + 12

3x = 12

x = 4 (tens digit)

2x = 8 (units digit)

**So,**

The number is (4 x 10) + 8 or 48.

- Eight times the sum of the digits of a certain two-digit number exceeds the number by 19.The tens digit is 3 more than the units digit. Find the number.

Solution

Let x = units digit (smaller)

x + 3 = tens digit

then the number is 10(x + 3) + x.

**Equation**

Eight times the sum of the digits exceeds the number by 19.

8[ x + (x + 3) ] – [ 10(x + 3) + x ] = 19

8[ 2x + 3 ] – [ 10x+30 + x ] = 19

16x + 24 – [ 9x + 30 ] = 19

16x + 24 – 9x – 30 = 19

5x – 6 = 19

5x = 25

x = 5 (units digit)

x + 3 = 8 (tens digit)

**So,**

The number is (8 x 10) + 5 or 85.

- The ratio of the units digit to the tens digit of a two-digit number one-half. The tens digit is 2 more than the units digit. Find the number.

Solution

Let x = units digit

x + 2 = tens digit

The ratio is a fractional relationship.

**Equation**

x/(x + 2) = 1/2

Multiply by the LCD, 2(x + 2)

x = 2 (units digit)

x + 2 = 4 (tens digit)

**Answer**

The number is (4 x 10) + 2 or 42.

- There is a two-digit number whose units digit is 6 less than the tens digit. Four times the tens digit plus five times the units digit equals 51. Find the digits.

Solution

Let x = units digit

x + 6 = tens digit

Four times the tens digit plus five times the units digit equal 51.

**Equation**

4(x + 6) + 5x = 51

4x + 24 + 5x = 51

9x + 24 = 51

9x = 27

x = 3 (units digit)

x + 6 = 9 (tens digit)

**Answer**

The number is (9 x 10) + 3 or 93.

- The tens digit is 2 less than the units digit. If the digits are reversed, the sum of the reversed number and the original number is 154. Find the original number.

Solution

Let x = units digit

x – 2 = tens digit

Then the number is 10(x – 2) + x and the reversed number is 10(x) + (x – 2)

**Equation**

The reversed number plus the original number equal 154.

10(x) + (x – 2) + 10(x – 2) + x = 154

10x + x – 2 + 10x -20 + x = 154

22x – 22 = 154

22x = 176

x = 8 (units digit)

x – 2 = 6 (tens digit)

**Answer**

The number is (6 X 10) + 8 or 68.

- A three-digit number has a tens digit 2 greater than the units digit and a hundreds digit 1 greater than the tens digit. The sum of the tens and hundreds digits is three times the units digit. What is the number?

Solution

Let x = units digit

x + 2 = tens digit

(x + 2) + 1 = hundreds digit

**Equation**

The sum of the tens and hundreds digits is three times the units digit.

(x + 2) + (x + 2) + 1 = 3x

2x + 5 = 3x

-x = -5

x = 5 (units digit)

x + 2 = 7 (tens digit)

(x + 2) + 1 = 8 (hundreds digit)

**Answer**

The number is (8 x 100) + (7 x 10) + 5 or 875.

- The sum of the digits of a two-digit number is 9. The value of the number is 12 times the tens digit. Find the number.

Solution

Let u = units digit

Let t = tens digit

Let u + 10t = the number itself

**Equation**

u + t = 9 sum of the digits is 9

u + 10t = 12t value of the number is 12 times the tens digit

u = 2t sub this into u + t = 9

2t = t = 9

3t = 9

t = 3

So the tens digit is 3 and the units must be 6

So the number is 36

Check

3 + 6 = 9

36 = 3(12)

36 = 36

- The sum of the digits of a two-digit number is 12. If 15 is added to the number, the result is 6 times the units digit. Find the number.

Solution

Let u = units digit

Let t = tens digit

Let u + 10t = the number itself

**Equation**

u + t = 12 sum of the digits is 12

u + 10t + 15 = 6u if 15 is added to the number, the result is 6 times the units digit

10t + 15 = 5u Simplify

2t + 3 = u divide through by 5

u = 2t + 3 sub this into u + t = 12

2t + 3 + t = 12

3t + 3 = 12

3t = 9

t = 3

So the tens digit is 3 and the units must be 9

So the number is 39

Check

3 + 9 = 12

39 + 15 = 6(9)

54 = 54

- The sum of the digits of a two-digit number is 8. If the digits of the number are reversed, the new number is 18 less than the original number. Find the number.

Solution

Let u = units digit

Let t = tens digit

Let u + 10t = the original number

10u + t = the number with the digits reversed

**Equation**

u + t = 8 sum of the digits is 8.

10u + t + 18 = u + 10t if the digits of the number are reversed, the new number is 18 less than the original number

9u + 18 = 9t simplify

u + 2 = t divide through by 9

t = u + 2 sub this into u + t = 8

u + (u + 2) = 8

2u + 2 = 8

2u = 6

u = 3

So the units digit is 3 and the tens must be 5

So the number is 53

Check

3 + 5 = 8

35 + 18 = 53

53 = 53

- The tens digit of a two-digit number is twice the units digit. If the digits are reversed, the new number is 36 less than the original number. Find the number.

Solution

Let u = units digit

Let t = tens digit

Let u + 10t = the original number

10u + t = the number with the digits reversed

**Equation**

t = 2u

10u + t + 36 = u + 10t

9u + 36 = 9t simplify

u + 4 = t divide through by 9

t = u + 4 sub this into t = 2u

2u = u + 4

u = 4

So the units digit is 4 and the tens must be 8

So the number is 48 and the number with the digits reversed is 84

Check

8 = 2(4)

48 + 36 = 84

- The units digit of a two-digit number is 4 times the tens digit. If the digits are reversed, the new number is 54 more than the original number. Find the number.

Solution

Let u = units digit

Let t = tens digit

Let u + 10t = the original number

10u + t = the number with the digits reversed

**Equation**

u = 4t

10u + t – 54 = u + 10t

9u – 54 = 9t simplify

u – 6 = t divide through by 9

u = t + 6 solve for u

u = t + 6 sub this into u = 4t

t + 6 = 4t

6 = 3t

t = 2

So the tens digit is 2 and the units digit must be 8

So the number is 28 and the number with the digits reversed is 82

Check

8 = 4(2)

82 = 28 + 54

- The sum of the digits of a two-digit number is 11. If 27 is added to the number, the digits will be reversed. Find the number.

Solution

Let u = units digit

Let t = tens digit

Let u + 10t = the original number

10u + t = the number with the digits reversed

**Equation**

u + t = 11

u + 10t + 27 = 10u + t

9t + 27 = 9u simplify

t + 3 = u divide through by 9

u = t + 3

u = t + 3 sub this into u + t = 11

(t + 3) + t = 11

2t + 3 = 11

2t = 8

t = 4

So the tens digit is 4 and the units digit must be 7

So the number is 47 and the number with the digits reversed is 74

Check

7 + 4 = 11

47 + 27 = 74

74 = 74

- The units digit of a two-digit number is 1 less than 3 times the tens digit. It the digits are reversed, the new number is 45 more than the original number. Find the number.

Solution

Let u = units digit

Let t = tens digit

Let u + 10t = the original number

10u + t = the number with the digits reversed

**Equation**

u + 1 = 3t

10u + t – 45 = u + 10t

9u – 45 = 9t simplify

u – 5 = t divide through by 9

u = t + 5 solve for u

u = t + 5 sub this into u + 1 = 3t

(t + 5) + 1 = 3t

t + 6 = 3t

2t = 6

t = 3

So the tens digit is 3 and the units digit must be 8

So the number is 38 and the number with the digits reversed is 83

Check

8 + 1 = 3(3)

38 + 45 = 83

83 = 83