Question

64. A two-digit number is obtained by either
multiplying the sum of the digits by 8 and then
subtracting 5 or by multiplying the difference of the
digits by 16 and then adding 3. Find the number,
[NCERT Exemplar)​

Answer 1

AnswEr :

Let the two digit number be 10x + y

◐ Two digit number is obtained by multiplying sum of the digits by 8 and subtracting 5

⇒ 10x + y = 8(x + y) - 5

⇒ 10x + y = 8x + 8y - 5

⇒ 10x - 8x + y - 8y + 5 = 0

⇒ 2x - 7y + 5 = 0⠀⠀⠀⠀--( ¡ )

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◐ Two digit number is obtained by multiplying the difference of the digits by 16 and adding 3

⇒10x + y = 16(x - y) + 3

⇒10x + y = 16x - 16y + 3

⇒10x - 16x + y + 16y - 3 = 0

⇒-6x + 17y - 3 = 0⠀⠀⠀⠀--( ¡¡ )

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━━━━━━━━━━━━━━━━━━━━━━━━

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Multiplying (¡) by 3 and Now ;

➟ 6x - 21y + 15 = 0

➟ -6x + 17y -⠀3 = 0

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➟ - 4y + 12 = 0

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➟ 12 = 4y

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➟ $\large\tt{\cancel\frac{12}{4}}$ = y

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➟ y = 3

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Putting the Value of y = 3 in (¡)

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➟ 2x - 7y + 5 = 0

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➟ 2x - (7 × 3) + 5 = 0

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➟ 2x - 21 + 5 = 0

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➟ 2x - 16 = 0

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➟ 2x = 16

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➟ x = $\large\tt{\cancel\frac{16}{2}}$

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➟ x = 8

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$\large\therefore$ Required Number is 83.

Answer 2

Answer :

Let the Number be a and b.

A.T.Q.

➡ 8(a + b) - 5 = 10a + b⠀⠀⠀---(1)

➡ 16(a - b) + 3 = 10a + b⠀⠀---(2)

★ From (1) and (2)

➡ 8(a + b) - 5 = 16(a - b) + 3

➡ 8a + 8b - 5 = 16a - 16b + 3

➡ 8a - 16a + 8b + 16b - 5 - 3 = 0

➡ - 8a + 24b - 8 = 0

➡ 24b = 8a + 8

➡ 8 × 3b = 8(a + 1)

• Cancelling 8 from Both Side

➡ 3b = a + 1

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★ Now We will try Trial and Error Method

➞ Let's take a = 2 and, b = 1

➞ 3b = a + 1

➞ 3 × 1 = 2 + 1

➞ 3 = 3

† Number can be 21

» 8(a + b) - 5 = 16(a - b) + 3

» 8(2 + 1) - 5 = 16(2 - 1) + 3

» 8 × 3 - 5 = 16 × 1 + 3

» 24 - 5 = 16 + 3

» 19 = 19 . Verified ✓

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➞ Let's take a = 8 and, b = 3

➞ 3b = a + 1

➞ 3 × 3 = 8 + 1

➞ 9 = 9

† Number can be 83 also.

» 8(a + b) - 5 = 16(a - b) + 3

» 8(8 + 3) - 5 = 16(8 - 3) + 3

» 8 × 11 - 5 = 16 × 5 + 3

» 88 - 5 = 80 + 3

» 83 = 83 . Verified ✓

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✡ That Number can be 21 or, 83 both.