... their ratio will be 2 : 11. How much does Terry have ?
Answers (2)
Let
T = Amount of Terry's money,
A = Amount of Aaron's money
before any financial transaction has taken place.
1st condition:
If Terry gives $100 to Aaron the ratio of their money will be 4 : 9
In mathematical terms:
(T - 100) / (A + 100) = 4/9 ; which gives us
Equation 1 :
9T - 900 = 4A + 400
2nd condition:
If Terry gives $300 to Aaron the ratio will be 2 : 11
(T - 300) / (A + 300) = 2/11 ; which gives us
Equation 2 :
11T - 3300 = 2A + 600
Subtract equation 1 from equation 2:
2T - 2400 = -2A + 200
2T - 2600 = -2A ; divide by -2, which yields
Equation 3 :
A = -T + 1300 ; As we can now express A in terms of T, this is our key to isolate T:
We substitute A by -T+1300 in, say, equation 1:
9T - 900 = 4(-T + 1300) + 400
9T - 900 = -4T +5200 +400
9T - 900 = -4T + 5600
13T = 6500
T = 500 ; here we are essentially done but in order to validate our result, ...
... we insert this value for T in Equation 3:
A = -500 + 1300
A = 800 ; and check, wether the results fit the primary conditions:
1)
(500 - 100) / (800 + 100) = 400 / 900 = 4/9 ;;TRUE
2)
(500 - 300) / (800 + 300) = 200 / 1100 = 2/11 ;;TRUE
So, Terry's got $500 and Aaron's got $800.
You don't say which is the top of the ratio and which is the bottom.
(A + 100) / (T - 100) = 4/9
(A + 300) / (T - 300) = 2/11
Reducing:
(A + 100) / (T - 100) = 4/9 Multiply by 9(T -100)
9(A + 100) = 4(T - 100) Evaluate.
9A + 900 = 4t - 400 Subtract 900.
9A = 4T - 1300 Divide by 9.
A = 4T/9 - 1300/9
Reduce the second equation:
(A + 300) / (T - 300) = 2/11 Multiply by 11(T - 300).
11(A + 300) = 2(T - 300)
11A + 3300 = 2T - 600 Subtract 3300.
11A = 2T - 3900 Divide by 11.
A = 2T/11 - 3900/11
Now you have two expressions equal to A, so just set them equal to each other.
4T/9 - 1300/9 = 2T/11 - 3900/11 Multiply by 99 to get rid of fractions.
44T - 14300 = 18T - 35100 Subtract 18T.
26T - 14300 = -35100 Add 14300.
26T = -20800 Divide by 26.
T = -800
Plug this into either original equation to get A.
(A + 100) / (T - 100) = 4/9
(A + 100)/(-900) = 4/9
A + 100 = -400
A = -500
Check:
(A + 300) / (T - 300) = 2/11
(-500 + 300)/(-800 - 300) = 2/11
-200/-1100 = 2/11
You might think that does not correspond to real life, but it is a valid answer. It illustrates the importance of correct presentation. Math is valid only when it describes reality. Mistrakes kill people.