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Answers
Answer:
The average age of A and B is 20 years. If A is replaced by C, the average age becomes 19 years and if B is replaced by C, the average age becomes 21 years. Find the ages of A, B and C.
Answer:
Ages of A, B & C are 22 years, 18 years and 20 years.
Explanation:
Given that:
Average age of A and B = 20 years
Average age of C and B = 19 years
Average age of A and C = 21 years
To Find:
Ages of A, B, & C?
Solution:
Let age of A be m years, age of B be n years and age of C be o years.
According to the question,
⇒ \sf Average\:age_{(A\:\&\:B)} = 20\:yearsAverageage
(A&B)
=20years
⇒ \sf \dfrac{m + n}{2} = 20
2
m+n
=20
⇒ \sf m + n = 20\:\times\:2m+n=20×2
➠ \bf\red{m + n = 40}\quad- - - (1)m+n=40−−−(1)
And,
⇒ \sf Average\:age_{(C\:\&\:B)} = 19\:yearsAverageage
(C&B)
=19years
⇒ \sf \dfrac{o + n}{2} = 19
2
o+n
=19
⇒ \sf o + n = 19\:\times\:2o+n=19×2
➠ \bf\green{o + n = 38}\quad- - - (2)o+n=38−−−(2)
Also,
⇒ \sf Average\:age_{(A\:\&\:C)} = 21\:yearsAverageage
(A&C)
=21years
⇒ \sf \dfrac{m + o}{2} = 21
2
m+o
=21
⇒ \sf m + o = 21\:\times\:2m+o=21×2
➠ \bf\purple{m + o = 42}\quad- - - (3)m+o=42−−−(3)
Adding (1), (2) & (3) we get,
⇒ \sf m + n + o + n + m + o = 40 + 38 + 42m+n+o+n+m+o=40+38+42
⇒ \sf m + m + n + n + o + o = 120m+m+n+n+o+o=120
⇒ \sf 2m + 2n + 2o = 1202m+2n+2o=120
⇒ \sf 2(m + n + o) = 1202(m+n+o)=120
⇒ \sf m + n + o = {\cancel{\dfrac{120}{2}}}m+n+o=
2
120
⇒ \sf m + n + o = 60m+n+o=60
From (1) put in above equation we get,
⇒ \sf 40 + o = 6040+o=60
⇒ \sf o = 60 - 40o=60−40
➠ \bf\pink{o = 20}o=20
Put o = 20 in (2) we get,
⇒ \sf 20 + n = 3820+n=38
⇒ \sf n = 38 - 20n=38−20
➠ \bf\blue{n = 18}n=18
Put n = 18 in (1) we get,
⇒ \sf m + 18 = 40m+18=40
⇒ \sf m = 40 - 18m=40−18
➠ \bf\orange{m = 22}m=22
Therefore,
✧ Age of A = m = 22 years
✧ Age of B = n = 18 years
✧ Age of C = o = 20 years
Let's Verify:
We know that,
↦ \sf\green{Average\:age_{(A\:\&\:B)} = 20\:years}Averageage
(A&B)
=20years
↦ \sf \dfrac{m + n}{2} = 20
2
m+n
=20
Put m = 22 & n = 18 in above equation we get,
↦ \sf \dfrac{22 + 18}{2} = 20
2
22+18
=20
↦ \sf {\cancel{\dfrac{40}{2}}} = 20
2
40
=20
↦ \sf 20 = 2020=20
➦ \bf\red{LHS = RHS}LHS=RHS
Hence, Verified ✔
And,
↦ \sf\pink{Average\:age_{(C\:\&\:B)} = 19\:years}Averageage
(C&B)
=19years
↦ \sf \dfrac{o + n}{2} = 19
2
o+n
=19
Put o = 20 & n = 18 in above equation we get,
↦ \sf \dfrac{20 + 18}{2} = 19
2
20+18
=19
↦ \sf {\cancel{\dfrac{38}{2}}} = 19
2
38
=19
↦ \sf 19 = 1919=19
➦ \bf\purple{LHS = RHS}LHS=RHS
Hence, Verified ✔
Also,
↦ \sf\orange{Average\:age_{(A\:\&\:C)} = 21\:years}Averageage
(A&C)
=21years
↦ \sf \dfrac{m + o}{2} = 21
2
m+o
=21
Put m = 22 & o = 20 in above equation we get,
↦ \sf \dfrac{22 + 20}{2} = 21
2
22+20
=21
↦ \sf {\cancel{\dfrac{42}{2}}} = 21
2
42
=21
↦ \sf 21 = 2121=21
➦ \bf\blue{LHS = RHS}LHS=RHS
Hence, Verified ✔
Therefore,
\red{\bigstar}★ Ages of A, B & C
\leadsto\:{\large{\boxed{\tt{\purple{22\:years,\:18\:years\:\&\:20\:years}}}}}⇝
22years,18years&20years
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