Tim’s weekly pocket money

Question

Tim’s weekly pocket money is $13.50 more than Sam’s. Sam spends $17.70 more than Tim every week. Tim spends 5/11 of his pocket money every week. Sam saves 1/2 as much as Tim every week. How much money does Sam spend in a week?

Answer 1

tim =sam +13.5 earnings

sam spends 11.7 more than tim

Tim spends 5/11 of his earnings

Sam saves 1/2 as much as tim

how much does Sam spend ?? 69.7 = $69..70

Tim spends 69.7 -11.7 = $58

Tim spends 5/11 of his earnings 58=5/11 x earnings. earnings =11/5 x 58 = 5638/5 =$112.76

Sam earns 112.76 -13.5= $99.25

Tim saves earnings minus spending = 112.76-58 =$54.76

Sam saves half as much = $27.38 = 99.25-58 =$40.47 uh oh, mistake(s) somewhere, recalculate this soon Or possible problem with the problem, misprinted or mis-copied. There may be an inconsistency in the problem that prevents a solution

there are 6 unknowns, each's earnings, spending & saving, and only 4 equations

that means an infinite number of solutions

each time you work the problem you may get another of those infinite solutions

Answer 2

Let's assume pocket money means money they have earned or been given each week and that each will likely spend less than they have in their pocket.

Let A = Tim's pocket money

B = Sam's pocket money

C = Tim spends

D = Sam spends

This means that Tim saves A - C per week (pocket minus spend equals saved).

So, Sam saves B - D per week.

We have four unknowns, let's see if given info gives us four equations.

A = B + 13.50 (Tim's pocket money is 13.5 more than Sam)

D = C + 17.70 (Sam spends 17.70 more than Tim); since we want to find D: C = D - 17.70)

C = (5/11)A (Tim spends 5/11 of his pocket money)

B - D = (1/2)(A - C) (Sam saves 1/2 as much as Tim).

Let's write the last two equations without fractions:

5A - 11C = 0

A - 2B - C + 2D = 0

Use the 1st two equations to replace A and C in those last two:

5(B + 13.50) - 11(D - 17.7) = 0

5B + 67.5 - 11D + 194.7 = 0

5B - 11D = -262.2

Then:

B + 13.50 - 2B - (D - 17.7) + 2D = 0

-B + D = -31.2

Down to two equations and two unknowns:

5B - 11D = -262.2

-B + D = -31.2

From the 2nd equation above: B = D + 31.2

Substitution into the 1st:

5(D + 31.2) - 11D = -262.2

5D + 156 - 11D = -262.2

-6D = -418.2

D = $69.70 (The amount Sam spends per week).

B = 69.7 + 31.2 = $100.90 (Sam's pocket money)

A = 100.90 + 13.50 = $114.40 (Tim's pocket money)

C = 69.70 - 17.70 = $52 (Tim spends)

Does Tim really spend 5/11 of his pocket money?

(5/11) 114.4 = 5(10.4) = $52

Of course there are many different ways to solve the system of 4 equations and 4 unknowns. You might try something different depending on what your class is working on: determinants, matrices, Gaussian Elimination...

And some would get fancier with the variables using something like T_p for Tim's pocket money...