Fractions and percentages

Question

I have a question for my sons homework that he and I are struggling to solve.

Janet spent 1/5 of her salary and an additional 146 on food every month. She then spent 1/3 of her remaining money and an additional 28 on transport. If she saved the remaining 1208, how much was her salary.

I tried solving this by going backwords: Initial = x 1/5 + 146 for food = x/5 +146 Remaining = 4/5x - 146

1/3 remaining + $28 on transport = 1/3 + 28 (4/5x - 146) = (4/15x - 48.6666 + 112/5x - 4088) = 22.6666x - 4136.666 = 1208 = 22.6666x = 5344.6666 x = 235

Which is obviously wrong. Any help would be greatly appreciated.

Answer 1

If $x$ is the salary, and if we assume the salary is all of the money there is to spend, then the entire salary $x$ can be partitioned as

$$x = \underbrace{(\tfrac15 x + 146)}_{\textrm{food}} + \underbrace{\tfrac13(\overbrace{x - (\tfrac15 x + 146)}^{\textrm{remaining after food}}) + 28}_{\textrm{transport}} + \underbrace{1208}_{\textrm{saved}}$$

Simplifying:

$$x = \tfrac15 x + 146 + \tfrac13x - \tfrac13(\tfrac15 x + 146) + 28 + 1208$$ $$x = \tfrac15 x + \tfrac13x - \tfrac1{15} x + 1382 - \tfrac{146}{3}$$ $$\tfrac{15}{15}x - \tfrac3{15}x - \tfrac5{15}x + \tfrac1{15}x = \tfrac{4146}{3} - \tfrac{146}{3}$$ $$\tfrac{8}{15}x = \tfrac{4000}{3}$$ $$x = \tfrac{15}{8}\cdot\tfrac{4000}{3} = \boxed{2500}$$ So the salary is $2500$.

Answer 2

Set $S=\text{salary}$ and partition in into $F=\text{food}$, $T=\text{transports}$ and $R=\text{remaining}$.

You know that $$ F=\frac{S}{5}+146, \quad R=1208, $$ and $$ T=\frac{1}{3}(S-F)+28. $$ Hence $$ T=\frac{1}{3}(S-\frac{S}{5}-146)+28=\frac{4}{15}S-\frac{32}{3}. $$

Lastly, since $F+T+R=S$, then $$ \left(\frac{S}{5}+146\right)+\left(\frac{4}{15}S-\frac{32}{3}\right)+1208=S. $$ Therefore $S=2500$.

Answer 3

Initial = x 1/5 + 146 for food = x/5 +146 Remaining = 4/5x - 146

This is really confusing. You say "Initial", but then you give the amount spent of food, and then you say "Remaining" without any line break.

1/3 + 28 (4/5x - 146) = (4/15x - 48.6666 + 112/5x - 4088)

You wrote "1/3 + 28 (4/5x - 146)", but then you calculated "(1/3 + 28) (4/5x - 146)". Those are different things, and both wrong.

Something that may have helped you in this case is if you had kept the units. 1/3 is a unitless quantity, but 28 has the unit of dollars. You can't add 1/3 and \$28, because they don't have the same units. If you instead have (1/3y +\$28), then now you're multiplying two dollars amounts, which gives you square dollars, which doesn't make any sense. You can't multiply \$28 and \$146. It's one third of the remainder plus \$28, not \$28 of the remainder plus one third. "Of" means times. So "1/3 of her remaining money and an additional 28 on transport." means "1/3 * (4x/5-\$146) + \$28".

I tried solving this by going backwords:

Except that you went through it forwards. If you were going backwards, you'd start at "1/3 of her remaining money and an additional 28 on transport". If x is the amount she stated with, and y is the amount she had after buying food, then y-y/3-28 = 1208. So 2y/3-28 = 1208, and 2y/3=1236, so y = 1872. You can then plug 1872 in to the first part to solve for x.

BTW, you can escape dollar signs by putting a backslash in front of them.

Answer 4

You don't say how old your son is or whether he's in a math course where they're starting to use variables like $x$. You can solve it without algebra by starting from the end and working backwards.

Janet saved the remaining 1208.
She spent $28$ on transportation before that, so she had $1208+28=1236$.
She spent $\frac13$ of the remaining money before that, so $1236$ is $\frac23$ of what she had, so she had $\frac32*1236=1854$.
She spent $146$ on food before that, so she had $1854+146=2000$.
Finally, she spent $\frac15$ of her salary before that, so $2000$ is $\frac45$ of her salary, so her salary is $\frac54*2000=2500$.

With problems like this, it's always a good idea to check your answer by plugging it into the original problem.

Let Janet's salary be $2500$, the result we calculated above.
Janet spent $\frac15$ of her salary ($\frac15$ of $2500$ is $500$) and an additional 146 on food every month. So she spent $500+146=646$ on food, leaving her with $2500-646=1854$.
She then spent $\frac13$ of her remaining money ($\frac13$ of $1854$ is $618$) and an additional 28 on transport. So she spent $618+28=646$ on transport, leaving her with $1854-646=1208$.
She saved the remaining $1208$.