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In a basket, the ratio of number of apples to the number of guavas is equal to the ratio of the number of guavas to the number of oranges. If the total number of fruits on the basket is 61, how many guavas are there?
A) 16
B) 20
C) 25
D) 28
E) 30
A) 16
B) 20
C) 25
D) 28
E) 30
17 Mar 2023, 06:31
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In a basket, the ratio of number of apples to the number of guavas is equal to the ratio of the number of guavas to the number of oranges. If the total number of fruits on the basket is 61, how many guavas are there?
Let a, g and o be the number of apples, guavas and oranges respectively. Also remember that a, g and o are Positive integers.
Given that \(\frac{a}{g} = \frac{g}{o}\) => \(a*o = g^2\)
Also, total number of fruits = a + g + o = 61
Substituting a = \(\frac{g^2}{o}\) in above equation, we get
\(\frac{g^2}{o} + g + o = 61\)
=> \(g^2 +g*o + o^2 = 61*o\)
=> \(o^2 +o(g-61) +g^2 = 0\)
Now we can substitute value of g from given options.
A. g = 16
\(o^2 - 45o + 256 = 0\)
In this case, the value of o is not an integer. So A is incorrect.
B. g = 20
\(o^2 - 41o + 400 = 0\)
On solving we get, o = 16 or 25. which means (a,o) is either (25,16) or (16,25)
So both of the sets will give \(\frac{a}{g} = \frac{g}{o}\).
Hence B is correct
C. g = 25
\(o^2 - 36o + 25^2 = 0\)
for values >= 25 the discriminant is imaginary. So C, D and E are out.
Answer B
Let a, g and o be the number of apples, guavas and oranges respectively. Also remember that a, g and o are Positive integers.
Given that \(\frac{a}{g} = \frac{g}{o}\) => \(a*o = g^2\)
Also, total number of fruits = a + g + o = 61
Substituting a = \(\frac{g^2}{o}\) in above equation, we get
\(\frac{g^2}{o} + g + o = 61\)
=> \(g^2 +g*o + o^2 = 61*o\)
=> \(o^2 +o(g-61) +g^2 = 0\)
Now we can substitute value of g from given options.
A. g = 16
\(o^2 - 45o + 256 = 0\)
In this case, the value of o is not an integer. So A is incorrect.
B. g = 20
\(o^2 - 41o + 400 = 0\)
On solving we get, o = 16 or 25. which means (a,o) is either (25,16) or (16,25)
So both of the sets will give \(\frac{a}{g} = \frac{g}{o}\).
Hence B is correct
C. g = 25
\(o^2 - 36o + 25^2 = 0\)
for values >= 25 the discriminant is imaginary. So C, D and E are out.
Answer B
10 Apr 2023, 12:34
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Can someone illustrate an easier way to solve this problem ?
