Question

The difference of two numbers is 14 and difference in their square is 448, Find the numbers.

Answer please. ​

Answer 1

Given: The Difference of two numbers is 14. & the difference of Square of these two numbers is 448.

Need to find: The numbers?

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❍ Let's say, the the numbers be x and y respectively.

Given that,

• Difference of these two numbers is 14.

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$:\implies\sf x - y = 14\qquad\qquad\quad\sf\Bigg\lgroup eq^{n}\;(1)\Bigg\rgroup$

Also,

• Difference of Square of these numbers is 448.

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$:\implies\sf x^2 - y^2 = 448$

$:\implies\sf (x + y) (x - y) = 448\qquad\qquad\quad\sf\Bigg\lgroup \Big(x^2 - y^2\Big) = \Big(x + y \Big) \Big(x - y \Big)\Bigg\rgroup$

$:\implies\sf x + y \times 14 = 448$

$:\implies\sf x + y = \cancel\dfrac{448}{14}$

$:\implies\sf x + y = 32\qquad\qquad\quad\sf\Bigg\lgroup eq^{n}\;(2)\Bigg\rgroup$

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⌑ From eqₙ ( I ) & eqₙ ( II ) :

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$\longrightarrow\sf x + \cancel{\;y} + x - \cancel{~ y}= 32 + 14$

$\longrightarrow\sf 2x = 46$

$\longrightarrow\sf x = \cancel\dfrac{46}{2}$

$\longrightarrow{\pmb{\underline{\boxed{\frak{x = 23}}}}}\;\bigstar$

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✇ Putting value of x in eqₙ ( II ) :

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$\longrightarrow\sf x + y = 32$

$\longrightarrow\sf 23 + y = 32$

$\longrightarrow\sf y = 32 - 23$

$\longrightarrow{\pmb{\underline{\boxed{\frak{y = 9}}}}}\;\bigstar$

$\therefore{\underline{\textsf{Hence, the numbers are \textbf{23} and \textbf{9} respectively.}}}$

Answer 2

Question:

• The difference of two numbers is 14 and difference in their square is 448, Find the numbers.

Answer:

• The numbers are 23 and 9.

Explanation:

Given that:

• The difference of two numbers is 14
• The difference in their square is 448

To Find:

• The numbers?

Solution:

★ Things to know before solving ::

• a² - b² = (a + b) ( a - b)

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• Let the numbers be m and n.

According to the question,

• Their difference is 14.

Therefore,

➻ $\boxed{\bf{m - n = 14}}$ – – – (1)

Also,

• Difference of their squares is 448.

Therefore,

➻ m² - n² = 448

➻ $\boxed{\bf{(m + n) (m - n) = 448}}$ – – – (2)

From (1) put in (2) we get,

➻ (m + n) × 14 = 448

➻ m + n = $\sf {\cancel{\dfrac{448}{14}}}$

➻ $\boxed{\bf{m + n = 32}}$ – – – (3)

Adding (1) and (3) we get,

➻ (m - n) + (m + n) = 14 + 32

➻ m - n + m + n = 46

➻ m + m + n - n = 46

➻ 2m = 46

➻ m = $\sf {\cancel{\dfrac{46}{2}}}$

➻ $\underline{\boxed{\bf{m = 23}}}$

Put m = 23 in (3) we get,

➻ 23 + n = 32

➻ n = 32 - 23

➻ $\underline{\boxed{\bf{n = 9}}}$

∴ Requires numbers are 23 and 9.

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