Question

half of the sum of two numbers is 34 and half of their difference is 6 . find the numbers​

Answer 1

Step-by-step explanation:

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Answer 2

ANSWER :

✪ If half of the sum of two numbers is 34 and half of their difference is 6, then the numbers are : 40 and 28.

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SOLUTION :

❖ Given :-

• Half of the sum of two numbers = 34

• Half of the difference of two numbers = 6

❖ To Find :-

• The numbers = ?

❖ Calculation :-

Suppose,

• The numbers are : x and y

➦ According to first condition,

• Half of (x + y) = 34

$\bigstar \: \: \: \: \sf{ \large{ \dfrac{x + y}{2} = 34}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{ \large{ \implies \: x + y = 68 \: \: - - - - - > (1)}}$

➦ According to second condition,

• Half of (x - y) = 6

$\bigstar \: \: \: \: \sf{ \large{ \dfrac{x - y}{2} = 6}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{ \large{ \implies \: x - y = 12 \: \: - - - - - > (2)}}$

➦ Solving eq. (1) and eq. (2),

$\sf{ \large{(1) + (2) \implies (x + y) + (x - y) = 68 + 12}} \\ \\ \: \: \: \sf{ \large{ \implies x + \cancel{y}+ x - \cancel{y} = 80}} \\ \\ \sf{ \large{ \implies 2x =80}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{ \large{ \implies x = \dfrac{80}{2}}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{ \large{ \therefore \: \: \underline{ \: x = 40 \: }}} \: \: \: \: \: \: \:$

And,

$\sf{ \large{(1) - (2) \implies (x + y) - (x - y) = 68 - 12}} \\ \\ \: \: \: \sf{ \large{ \implies \cancel{x}+ y - \cancel{x} + y= 56}} \\ \\ \sf{ \large{ \implies 2y =56}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{ \large{ \implies y = \dfrac{56}{2}}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{ \large{ \therefore \: \: \underline{ \: y = 28 \: }}} \: \: \: \: \: \: \: \: \:$

• x = 40 and y = 28

✪ Hence, the numbers are : 40 and 28.

$\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }$

VERIFICATION :

We have,

• The numbers are : x = 40 and y = 28.

➦ According to first condition,

• Half of the sum of the numbers = 34

Or,

• Half of (x + y) = 34

$\: \: \bigstar \: \: \sf{Half \: \: of \: \: (x + y) = \dfrac{x + y}{2} } \: \: \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{ = \dfrac{40 + 28}{2}} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{= \dfrac{68}{2}} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{= 34}$

➦ According to second condition,

• Half of the difference of the numbers = 34

Or,

• Half of (x - y) = 34

$\: \: \bigstar \: \: \sf{Half \: \: of \: \: (x - y) = \dfrac{x - y}{2} } \: \: \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{ = \dfrac{40 - 28}{2}} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{= \dfrac{12}{2}} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{= 6}$

✪ Hence verified.