✒️NUMBERS
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[tex] \large\underline{\mathbb{PROBLEM}:} [/tex]
- The sum of two numbers is 44. the smaller number is five sixths of the larger number. What are the numbers?
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[tex] \large\underline{\mathbb{ANSWER}:} [/tex]
[tex] \qquad \LARGE \:\:\rm{24 \: and \: 20} [/tex]
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[tex] \large\underline{\mathbb{SOLUTION}:} [/tex]
» Let x and y be the smaller and the larger number respectively. create two equations by the given statements.
- [tex] \begin{cases} x + y = 44 \\ y = \frac56x \end{cases} \quad \begin{align} \tt{(eq. \: 1)} \\ \tt{(eq. \: 2)} \end{align} [/tex]
» Find x in the first equation then substitute it to the second equation in terms of y.
- [tex] \begin{cases} x = 44 - y \\ y = \frac56x \end{cases} [/tex]
- [tex] \begin{cases} x = 44 - y \\ y = \frac56(44-y) \end{cases} [/tex]
- [tex] \begin{cases} x = 44 - y \\ y = \frac{220 - 5y}6 \end{cases} [/tex]
- [tex] \begin{cases} x = 44 - y \\ y(6) = \frac{220 - 5y}6(6) \end{cases} [/tex]
- [tex] \begin{cases} x = 44 - y \\ 6y = 220 - 5y \end{cases} [/tex]
- [tex] \begin{cases} x = 44 - y \\ 6y + 5y = 220 \end{cases} [/tex]
- [tex] \begin{cases} x = 44 - y \\ 11y = 220 \end{cases} [/tex]
- [tex] \begin{cases} x = 44 - y \\ 11y/11 = 220 /11 \end{cases} [/tex]
- [tex] \begin{cases} x = 44 - y \\ y = 20 \end{cases} [/tex]
» Thus, the smaller number is 20. Substitute the value of y to the first equation to find x.
- [tex] \begin{cases} x = 44 - 20 \\ y = 20 \end{cases} [/tex]
- [tex] \begin{cases} x = 24 \\ y = 20 \end{cases} [/tex]
[tex] \therefore [/tex] The two numbers are 24 and 20.
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