... 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.