Question

Raman has three times as much money as Kamal. If Raman gives Rs. 750 to Kamal, then Kamal will have twice as much as left with Raman. How much had each originally?

Answer 1

Hint: This question is to be approached by appropriate representation of given details using variables and forming equations as per the data. As here we have two quantities of money to deal with, we have to define each with a variable. Then solving the formed equations to obtain these variable values will result in the needed solution.

Complete step-by-step answer:
Step 1: Our first work is to represent the given data using variables. As we have two inputs which are the money Raman originally had and money of Kamal he had originally, let X and Y be their representation respectively. That is,
X = Money with Raman initially
Y = Money with Kamal initially
Step 2: Now let us form equations regarding the data provided. Initially Raman had three times as much money as Kamal. Implies,
\[{\mathbf{X}} = {\mathbf{3Y}}\;\;\] … Formula 1
Step 3: As Raman gives Rs. 750 to Kamal, Y gets added by 750 which is given to be equal to two times money left with Raman which Raman’s get subtracted by 750 and is multiplied by 2. Thus,
\[{\mathbf{Y}} + {\mathbf{750}} = {\mathbf{2}}\left( {{\mathbf{X}} - {\mathbf{750}}} \right)\;\;\] … Formula 2
Simplifying Formula 2 we get,
\[Y + 750 = 2X - 1500\]
$2X = Y + 750 + 1500$
$2X = Y + 2250$
Step 4: Solving Formula 1 and final form of Formula 2, by subtracting Formula 1 from 3 times final form of formula 2 which gives LHS, 6X-X=5X and RHS, 6750 That is,
5X=6750 and
$X = \dfrac{{6750}}{5} = 1350$
Step 5: Now we can find Y from X value using Formula 1,
$Y = \dfrac{X}{3} = \dfrac{{1350}}{3} = 450$

Raman originally had Rs. 1350 and Kamal had Rs. 450.

Note: Solving the formed equations is the part where most of the students make errors. Solving can be made simpler if we concentrate on a single variable and try to find its value. As the other variable is dependent on the first one, we can easily obtain the value of the second. In the given solution we simplified it with respect to the variable X. The other way of doing it will be based on Y. Thus the simplification will be done by, Final form of Formula 2 -2(Formula 1). Then we obtain,
$0 = - 5Y + 2250$
$
  5Y = 2250 \\
  Y = 450 \\
 $
Then $X = 3Y = 1350$ . which is the same as the explained solution.