Question

A and B have 460 coins altogether. If \[\dfrac{3}{4}\] of $A$'s number of coins is equal to \[\dfrac{2}{5}\] of $B$'s number of coins, find how many coins must B give to A, so that they both have equal number of coins.

Answer 1

Hint: First we take a variable as $A$'s number of coins and another for $B$'s number of coins.
Then applying the two conditions we will get two equations.
By solving the equations we will get $A$'s number of coins and $B$'s number of coins.

Complete step by step answer:
It is given that A and B have 460 coins altogether.
Let, $A$'s number of coins be $x$ & $B$'s number of coins be $y$.
Again, the number of total coins is 460.
Thus, \[x + y{\rm{ }} = 460\]
Rearrange the equation and mark it as (1)
\[y = 460{\rm{ }} - {\rm{ }}x\] …… (1)
Again it is given that \[\dfrac{3}{4}\] of $A$'s number of coins is equal to \[\dfrac{2}{5}\] of $B$'s number of coins
That is \[\dfrac{{3x}}{4} = \dfrac{{2y}}{5}\]
By cross multiplication of the above equation we have,
\[3x \times 5 = 2y \times 4\\\]
On simplifying the above equation, we get,
\[15x = 8y\]
Rearrange the equation and mark it as (2) we get,
\[x = \dfrac{{8y}}{{15}}.....(2)\]
Let us put the value of x from (2) in (1) we get
\[\dfrac{{8y}}{{15}} + y = 460\]
Let us simplify it in the left hand side we get,
\[\dfrac{{8y + 15y}}{{15}} = 460\]
\[23y = 460 \times 15\]
Let us solve the above equation for \[y\]we get,
 \[y = \dfrac{{460 \times 15}}{{23}}\]
\[y = 20 \times 15 = 300\] On simplifying the above equation,
\[y = 300\]
Substitute \[y\]in equation (1) we get,
\[\;x = 460 - 300 = 160.\]
Therefore, we have found \[\;x = 160,{\rm{ }}y = 300\]
Hence $A$'s number of coins & $B$'s number of coins are 160 & 300 respectively.
So, B has \[\left( {300 - 160} \right){\rm{ }} = 140\] more coins than A.
If we distribute these 140 coins equally between A and B then the number of coins of A and B will be the same.
Then B has to give A \[\dfrac{{140}}{2} = 70\] coins, to make the number of coins equal.

Hence, B must give A 70 coins, so that they both have equal numbers of coins.

Note:
In these kinds of questions, the logic lies in the given information. Grab that logic to form equations. If we have two equations with two variables then by method of substitution we get the solution. To find who has got more number of coins we should subtract the total number of coins each have.