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If m, 10, n, 40 are in continued proportion then find the positive values of m and n.
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Verified answer
Answer:
[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\sf \: m=5\qquad \: \\ \\& \qquad \:\sf \: n = 20\end{aligned}} \qquad \: \\ \\ [/tex]
Step-by-step explanation:
Given that,
[tex]\sf \: m, 10, n, 40 \: are \: in \: continued \: proportion. \\ \\ [/tex]
[tex]\sf \: \dfrac{m}{10} = \dfrac{10}{n} = \dfrac{n}{40} \: \\ \\ [/tex]
Taking second and third member, we get
[tex]\sf \: \dfrac{10}{n} = \dfrac{n}{40} \: \\ \\ [/tex]
[tex]\sf \: {n}^{2} = 40 \times 10 \\ \\ [/tex]
[tex]\sf \: {n}^{2} = 10 \times 10 \times 2 \times 2 \\ \\ [/tex]
[tex]\sf \: {n}^{2} = {20}^{2} \\ \\ [/tex]
[tex]\sf\implies \bf \: n = 20 \\ \\ [/tex]
Now, Taking first and second member, we get
[tex]\sf \: \dfrac{m}{10} = \dfrac{10}{n} \: \\ \\ [/tex]
On substituting the value of n, we get
[tex]\sf \: \dfrac{m}{10} = \dfrac{10}{20} \: \\ \\ [/tex]
[tex]\sf \: \dfrac{m}{10} = \dfrac{1}{2} \: \\ \\ [/tex]
[tex]\sf \: m = \dfrac{10}{2} \: \\ \\ [/tex]
[tex]\sf\implies \bf \: m = 5 \\ \\ [/tex]
Hence,
[tex] \: \: \sf\implies \:\boxed{\begin{aligned}& \qquad \:\bf \: m=5\qquad \: \\ \\& \qquad \:\bf \: n = 20\end{aligned}} \qquad \: \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
1. Alternendo
[tex]\sf \:If\:\dfrac{a}{b} = \dfrac{c}{d} ,\:then \: \bf \: \dfrac{a}{c} = \dfrac{b}{d} \\ \\ [/tex]
2. Invertendo
[tex]\sf \:If\:\dfrac{a}{b} = \dfrac{c}{d} ,\:then \: \bf \: \dfrac{b}{a} = \dfrac{d}{c} \\ \\ [/tex]
3. Componendo
[tex]\sf \:If\:\dfrac{a}{b} = \dfrac{c}{d} ,\:then \: \bf \: \dfrac{a + b}{b} = \dfrac{c + d}{d} \\ \\ [/tex]
4. Dividendo
[tex]\sf \:If\:\dfrac{a}{b} = \dfrac{c}{d} ,\:then \: \bf \: \dfrac{a - b}{b} = \dfrac{c - d}{d} \\ \\ [/tex]
[tex] \rm \: the \: positive \: value \: of \: m \: and \: n \: are : - \\ \\ \underline { \boxed { \sf \red{\: n \: = \: 20}}} \\ \\ \underline { \boxed { \sf \red{\: m \: = \: 5}}} \\ [/tex]
Step-by-step explanation:
[tex]\tt \large\: {\red{ \underline {question : }}} \\ [/tex]
If m, 10, n, 40 are in continued proportion then find the positive values of m and n.
[tex]\tt \large \: {\pink{ \underline {given : }}} \\ [/tex]
[tex] \\ \tt \large \: {\green{ \underline {to \: find : }}} \\ [/tex]
[tex] \\ \tt \large \: {\green{ \underline {solution: }}} \\ \\ \sf \: as \: given - - \\ \sf\: m \: 10 \: n \: 40 \: are \: in \: continued \: proportion \: \\ \\ \sf \: so \: \: \: \: \dfrac{m}{10} \: = \: \dfrac{10}{n} \: = \: \dfrac{n}{40} \\ \\ \sf \: on \: comparing \: \: \: \frac{10}{n} = \frac{n}{40} \\ \\ \sf \:n \: \times \: n = 10 \: \times \: 40 \\ \\ \sf {(n)}^{2} \: = \: 400 \\ \\ \bf \: therefore \\ \\ \underline{ \boxed{ \sf n = 20}} \\ \\ \underline{ \bf \: now \: on \: comparing : - } \\ \\ \sf \dfrac{m}{10} \: = \: \dfrac{10}{n} \\ \\ \sf \: m \: \times \: n \: = \: 10 \: \times \: 10 \\ \\ \sf \: mn \: = \: 100 \\ \\ \underline{ \bf \: put \: the \: value \: of \: n \: in \: above \: - } \\ \\ \sf \: m \: \times \: (20) \: = \: 100 \\ \\ \sf \: m \: = \: \dfrac{100}{20} \\ \\ \underline{ \boxed{ \sf m \: = \: 5 }}\\ [/tex]
[tex] \bf \: therefore \: the \: positive \: value \: of \: m \: and \: n \: are : - \\ \\ \underline { \boxed { \sf \red{\: n \: = \: 20}}} \\ \\ \underline { \boxed { \sf \red{\: m \: = \: 5}}} \\ \\ [/tex]