10. A dealer bought two cars. He sold the first one for $40 \%$ more than he paid for it and the second one for $60 \%$ more than he paid for it. The total sum he received for the two cars was $54 \%$ more that the total sum he paid for them. When written in its lowest terms, the ratio of the prices the dealer paid for the first and the second car was $a: b$. What is the value of $11 a+13 b$ ?
SOLUTION
124

Let the prices the dealer paid for the first and second cars be $X$ and $Y$ respectively. He sells the cars for $1.4 X$ and $1.6 Y$.
Since he received $54 \%$ more than he paid, we have $1.54(X+Y)=1.4 X+1.6 Y$.
Therefore, $0.14 X=0.06 Y$ which gives us the ratio $X: Y=6: 14=3: 7$.
Therefore $a=3$ and $b=7$ and $11 a+13 b=11 \times 3+13 \times 7=33+91=124$.