Question

If three times the larger of two numbers is divided by the smaller, we get 4 as the quotient and 8 as the remainder. If five times the smaller is divided by the larger, we get 3 as the quotient and 5 as the remainder. Find the numbers. (Ans.20,13)

Answer 1

let larger number be x

and smaller number be y

First condition

Dividend = divsor * quotient + remainder

3x= y*4+3

3x=4y+3

3x-4y=3

second condition

7y=5x+1

7y-5x=1

3 x -4 y = 3 .............1

-5 x + 7 y = 1 .............2

Eliminate y

multiply (1)by 7

Multiply (2) by 4

21 x -28 y = 21

-20 x + 28 y = 4

Add the two equations

1 x = 25

/ 1

x = 25

plug value of x in (1)

3 x -4 y = 3

75 -4 y = 3

-4 y = 3 -75

-4 y = -72

y = 18 1

x= 25

y= 18

Answer 2

$\huge\underline\mathrm\orange{SOLUTION:-}$

Given:

• If three times the larger of two numbers is divided by the smaller then we get 4 as the quotient and 8 as the remainder.

• If Five years times the smaller is divided by the larger then we get 3 as the quotient and 5 as the remainder.

Need To Find:

• The Numbers = ?

$\underline\mathtt\green{EXPLANATION:-}$

Suppose the First Number(Larger Number) be a

And, Suppose the Second Number(Smaller Number) be b

Therefore:

$\boxed{\underline{\sf{We \ Know \ that :}}}$

$\boxed{\sf{\pink{Divided = Divisor \times Quotient + Remainder }}}$

According to the First Condition:

• If three times the larger of two numbers is divided by the smaller then we get 4 as the quotient and 8 as the remainder.

$\implies \sf{3a = b \times 4 + 8}$

$\implies \sf{3a = 4b + 8}$

$\implies \sf{a = \dfrac{4b + 8}{3}}$ ____(1) Equation

According to the Second Condition:

• If Five years times the smaller is divided by the larger then we get 3 as the quotient and 5 as the remainder.

$\implies \sf{5b = a \times 3 + 5}$

$\implies \sf{5b = 3a + 5}$

║Now Put the Value of a From the Equation First ║

$\implies \sf{5b = 3(\dfrac{4b + 8}{3} ) + 5}$

$\implies \sf{5b = \dfrac{12b + 24}{3} + 5}$

$\implies \sf{5b = \dfrac{12b + 24 + 15}{3}}$

$\implies \sf{15b = 12b + 39}$

$\implies \sf{15b - 12b = 39}$

$\implies \sf{3b = 39}$

$\implies \sf{b = \dfrac{39}{3}}$

$\implies \boxed{\sf{b = 13}}$

║Now Put the Value of b in First Equation ║

$\implies \sf{a = \dfrac{4b + 8}{3}}$

$\implies \sf{a = \dfrac{4(13) + 8}{3}}$

$\implies \sf{a = \dfrac{52 + 8}{3}}$

$\implies \sf{a = \dfrac{60}{3}}$

$\implies \boxed{\sf{a = 20}}$

Therefore:

$\boxed{\large{\sf{\red{The \ Larger \ Number = a = 20}}}}$

$\boxed{\larger{\sf{\red{The \ Smaller \ Number = b = 13}}}}$

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