how much money did amy spend

Question

amy and pamela went shopping together with a total sum of $532. amy spent half as much as pamela .the amount amy had left was $165 more then the hat she had spent. she had thrice as much money left as pamela.

Answer 1

3A + (A + 165) + \tfrac{1}{3}(A+165) = 532

4A + 165 + \tfrac{1}{3}A + 55 = 532

\left(4A + \tfrac{1}{3}A\right) + 220 = 532

\tfrac{13}{3}A + 220 = 532

\tfrac{13}{3}A = 312

A = \tfrac{312 \times 3}{13} = 72

答案

Amy 花了 $72。

Pamela 花了 $144，

Amy 剩下 $237，

Pamela 剩下 $79。

检查：72 + 144 + 237 + 79 = 532 ✅ 条件完全符合。

Answer 2

This can be solved with a system of several equations, making one equation from each piece of information given in the prompt, and adding a couple based on knowledge of how money works. There will be a total of 6 variables so we need 6 equations to be able to solve this.

Start = O, Spent = S, Final = F

"amy and pamela went shopping together with a total sum of $532" --> 1. A_O + P_O = 532

"amy spent half as much as pamela" --> 2. A_S = 1/2*P_S

"the amount amy had left was $165 more than what she had spent" --> 3. A_F = A_F + 165

"she [Amy] had thrice as much money left as pamela" --> 4. A_F = 3*P_F

We also know that whatever they have left must be the difference between what they started with and what they spent, so we get: 5. A_F = A_O - A_S and 6. P_F = P_O - P_S

Now to start to solve, we have to decide which variable we are going to isolate first. I chose A_F

Substitute Equation 5 into 3: A_F = A_O - A_F + 165

Rearrange and substitute Equation 1: 2*A_F = 532 - P_O + 165

Add and substitute Equation 6: 2*A_F= 697 - (P_S+P_F)

Factoring the negative and substitute Equations 2 and 4: 2*A_F = 697 - 2*A_S - 1/3*A_F

Rearrange and substitute Equation 3: 2 1/3*A_F = 697 - 2(A_F-165)

Now we solve:

2 1/3*A_F = 697 - 2*A_F + 330

4 1/3*A_F = 1027

A_F = 237



Now solve the rest:

A_S = A_F - 165 = 237 - 165 = 72

A_O = A_S + A_F = 237 + 72 = 309

P_F = 1/3*A_F = 1/3*237 = 79

P_S = 2*A_S = 2*72 = 144

P_O = P_S + P_F = 79 + 144 = 223

Now for a final check:

A_start + P_start = 532

309 + 223 = 532

532 = 532

Answer 3

Amy spent $237 = half as much as pam spent

leaving Amy with 165 more than she spent

amy had 3 times as much left as Pam had left

together they had $532

Amy spent $237, Pam spent $144

Amy had $237 left, Pam had $79 left

those numbers work, exactly