Question

the sum of two numbers is 64 one of the number is 8 less than the other number find the numbers​

Answer 1

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \purple{ \: \: SOLUTION \: \: } \mid}}}}}$

$\underline{ \mathfrak{ \: \: Given \: \: }} \\ \\ \text{ \pink{The sum of the two numbers = 64}} \\ \text{ \pink{One number is less than the another number.}} \\ \\ \underline{ \mathfrak{ \: \: To \: \: find : \to \: }} \\ \\ \: \: \: \: \: \: \: \: \: \text{ \blue{The numbers = ?}} \\ \\ \underline{ \mathfrak{ \: \: Suppose, \: \: }} \\ \\ \text{ \red{The bigger number is = x }} \\ \text{ \red{The smaller number is = y}}$

$\underline {\bold{ \: \: A.T.Q., \: \: }} \\ \\ \: \: \: \: \: \: \: \: \: \: \tt{ \red{x + y = 64} \: \: - - - - - > (1)} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \tt{x - 8 = y } \\ \tt{ \implies \: \red{ x - y = 8} \: \: - - - - - (2)}$

$\tt{ \therefore \: (1) + (2) \implies \: 2x = 72} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt{ \implies \:x = \frac{72}{2} } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt{ \therefore \: \red{ x = 36}}$

$\underline{ \mathfrak{ Again, \: }}\\ \\ \tt{ \therefore \: (1) - (2) \implies \: 2y = 56} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt{ \implies \:y = \frac{56}{2} } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt{ \therefore \: \red{ y = 28}}$

$\sf{ \therefore \: \: The \: \: numbers \: \: are : \: \: \red{36} \: and \: \red{ 28}.}$

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \purple{ \: \: VERIFICATION \: \: } \mid}}}}}$

$\underline{ \mathfrak{ \: \: We \: \: have, \: \: }} \\ \\ \text{ \red{The bigger number = 36}} \\ \text{ \red{The smaller number = 28}} \\ \\ \therefore \: \text{Sum of the numbers =( \red{36 +28 }}) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \text{ = \red{ 64}} \\ \\ \underline{ \mathfrak{ Again, \: }}\\ \\ \text{Bigger number - 8 = \red{36 - 8}} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \text{= \red{28}} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \text{ = Smaller number} \\ \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underline{ \bold{ \: \: Hence \: \: verified. \: \: }}$

Answer 2

Answer:

$\large\star\underline\textsf{ The number are : 36 and 28 }$

$\huge\underline\textsf{Explantion:- }$

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$\</u><u>star</u><u>\underline\textsf{</u>s<u>u</u>p<u>p</u><u>o</u><u>s</u><u>e</u><u>:- }$

$\star\underline\textsf{The bigger number is = \red A}$

$\star\underline\textsf{The smaller number is = \red B}$

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$\large\star\underline\textsf{Using Identities :- }$

$\boxed{\bf(A + B) { }^{2} = A {}^{2} + 2AB + B {}^{2}}$

$\boxed{\bf(A - B) {}^{2} = a {}^{2} - 2AB + B {}^{2}}$

$\large\star\underline\textsf{A.T.Q, }$

$\leadsto\sf A + B = 64 - - - - > 1.identity$

$\leadsto\sf A - 8 = B$

$\leadsto\sf A - B = 8 - - - - > 2.identity$

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$\leadsto\sf(1) + (2) = 2a = 72$

$\leadsto\sf A = \frac{72}{2}$

$\leadsto\bf A = 36$

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$\star\underline\textsf{Now do again with 2.identity}$

$\leadsto\sf(1) - (2) = 2B = 56$

$\leadsto\sf B = \frac{56}{2}$

$\leadsto\sf B = 28$

$\large\star\underline\textsf{ The number are : \red3\red6 and \orange2\orange8 }$

$\huge\underline\textsf{Verification:- }$

$\small\underline\textsf{We have }$

$\bullet\textsf{ the bigger number=36}$

$\bullet\textsf{ the smaller number=28}$

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$\therefore\textsf{sum of number=[36+28]}$

$\textsf{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = 64}$

$\star\underline\textsf{now again, }$

$\small\underline\textsf{bigger number -8 = 36-8}$

$\textsf{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = 28}$

$\textsf{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = smaller number }$

$\huge\underline{</p><p>{\mathbb{\purple{HENCE\:VERIFIED. }}}}$

$\quad\boxed{\begin{minipage}{15em}{\large\underline\textbf{Some identitys }}} \\ \\</p><p></p><p>(A+B)^2=a^2+2ab+b^2 — — — —> 1.identity\\ </p><p></p><p>(a-b) ^2=a^2=a^2-2ab+b^2 — — — —>2.identity\\</p><p>(a+b)(a-b) =a^2-b^2— — — —>3.identity \\</p><p></p><p>\end{minipage}}$