Question
Mathematics Question on Linear Equations in two variables
The monthly incomes of A and B are in the ratio 8 : 7 and their expenditures are in the ratio 19 : 16. If each saves Rs 2500 per month, find the monthly income of each
Step 1: Let the incomes and expenditures be:
Income of A=8x,Income of B=7x.
Expenditure of A=19y,Expenditure of B=16y.
Step 2: Use the savings equation \textit{Savings = Income − Expenditure}:
8x−19y=2500(for A),
7x−16y=2500(for B).
Step 3: Solve the equations. From the first equation:
8x=19y+2500⟹x=819y+2500.
Substitute into the second equation:
7(819y+2500)−16y=2500.
Simplify:
8133y+17500−16y=2500.
Multiply through by 8:
133y+17500−128y=20000.
5y=2500⟹y=500.
Step 4: Find x:
x=819(500)+2500=89500+2500=812000=1500.
Step 5: Find incomes:
Income of A=8x=8(1500)=12000.
Income of B=7x=7(1500)=10500.
Correct Answer: Incomes are 12000 and 10500.
Solution
Step 1: Let the incomes and expenditures be:
Income of A=8x,Income of B=7x.
Expenditure of A=19y,Expenditure of B=16y.
Step 2: Use the savings equation \textit{Savings = Income − Expenditure}:
8x−19y=2500(for A),
7x−16y=2500(for B).
Step 3: Solve the equations. From the first equation:
8x=19y+2500⟹x=819y+2500.
Substitute into the second equation:
7(819y+2500)−16y=2500.
Simplify:
8133y+17500−16y=2500.
Multiply through by 8:
133y+17500−128y=20000.
5y=2500⟹y=500.
Step 4: Find x:
x=819(500)+2500=89500+2500=812000=1500.
Step 5: Find incomes:
Income of A=8x=8(1500)=12000.
Income of B=7x=7(1500)=10500.
Correct Answer: Incomes are 12000 and 10500.