Ten years ago the ratio between the ages of Mohan and Suman was 3:5. 11 years hence it will be 11:16. What is the present age of Mohan?
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Ten years ago the ratio between the ages of Mohan and Suman was 3:5. 11 years hence it will be 11:16. What is the present age of Mohan?
Let the present ages of Mohan and Suman be x and y respectively.
Therefore, 10 years ago their ages were:
Mohan = x - 10
Suman = y - 10
Therefore, 11 years from now there ages will be:
Mohan = x + 11
Suman = y + 11
Therefore, we can make two equations from the above information. These can then be solved according to the rules of quadratic equations.
We know that ten years ago the ratio between the ages of Mohan and Suman was = 3/5
Therefore,
(x - 10)/(y - 10) = 3/5
5(x - 10) = 3(y - 10)
5x - 50 = 3y - 30
5x - 3y = 20 ---------- Equation (1)
We also know that 11 years from now the ration between the ages of Mohan and Suman will be = 11/16
(x + 11)/(y + 11) = 11/16
16(x + 11) = 11(y +11)
16x + 176 = 11y + 121
16x - 11y = 55 ---------Equation (2)
Solving the two equations:
5x - 3y = 20 ------------- Equation (1)
16x - 11y = 55 ---------- Equation (2)
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In order to cancel out one of the variables x or y we need to multiply both the equations such that the values of the variables x or y equal out in both and one can be subtracted from the other to get a resulting value of 0.
In this case we are going to multiply equation (1) with 11 and equation (2) with 3 and then subtract equation (2) from equation (1)
55x - 33y = 220
-48x + 33y = -165
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7x - 0 = 55
7x = 55
x = 55/7
x = 7 6/7
Now substitute the value of x in equation (1)
5x - 3y = 20
5 (55/7) - 3y = 20
275/7 - 3y = 20
3y = 275/7 - 20
3y = (275 - 140)/7
y = 135/(7 x 3)
y = 135/21
y = 45/7
y = 6 3/7
Therefore we can check the values by substituting the answers in the two ratios above.
(x - 10)/(y-10) = 3/5
(55/7 - 10)/ (45/7 - 10) = 3/5
[(55 - 70)/7]/[(45 - 70)/7] = 3/5
(15/7)/(25/7) = 3/5
15/7 x 7/25 = 3/5
15/25 = 3/5
3/5 = 3/5
We can double check the values by substituting them in the second ration as well.
(x + 11)/(y + 11) = 11/16
(55/7 + 11)/(45/7 + 11) = 11/16
[(55 - 77)/7]/ [(45 - 77)/7] = 11/16
(22/7)/ (32/7) = 11/16
22/7 x 7/32 = 11/16
22/32 = 11/16
11/16 = 11/16
Therefore we know that the present age of Mohan is 55/7 which is equal to 7 6/7 years.
Similarly, Suman's present age is equal to 45/7 which is equal to 6 3/7 years.
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let present age of mohan = x
let present age of Suman = y
TEN YEARS AGO
age of mohan was = x - 10
age of suman was = y - 10
given that (x-10) / (y-10) = 3/5
⇒ 5(x -10) = 3(y -10)
⇒ 5x - 50 = 3y - 30
⇒ 5x - 3y = 20 ------------------------------ eq 1
11 years Hence
Age of mohan will = x + 11
Age of suman will = y + 11
given that (x + 11 ) / ( y + 11) = 11/16
⇒ 16(x + 11) = 11(y + 11)
⇒ 16x + 176 = 11y + 121
⇒ 16x - 11y = -55 -----------------------------------eq 2
now we have two equations to solve
5x - 3y = 20 ------------------------------ eq 1
16x - 11y = -55 --------------------------- eq 2
multiply eq 1 by 11 and eq 2 by 3
55x - 33y = 220 ------------------------------ eq 3
48x - 33y = -165 -------------------------------eq 4
subtraction eq 4 from eq 3
55x - 48x -33y + 33y = 220 - ( -165)
⇒ 7x = 385
⇒ x = 55
substituting value of x in equation 1
5 × 55 - 3y = 20
⇒ 3y = 275 - 20
⇒ 3y = 255
⇒ y = 255/3 = 85
Present age of mohan = x = 55 years