Emma works at a clothing store, and works on commission.

The amount she makes each week depends on how much clothing she sells, and she also receives a weekly amount regardless of her sales.
Emma also has to pay for her meals during her shifts, which detracts from her income.

Emma’s income can be modelled by the following equation:         I=100+12J+5T+7S+5H-8M

where
I is her income,
J is the number of jeans she sells,
T is the number of T-shirts she sells,
S is the number of shorts she sells,
H is the number of hats she sells, and
M is the number of meals she buys.

If Emma makes
$205 in a week and sells:
6 pairs of jeans,
4 shorts,  and
5 hats,
and buys 5 of her meals.
How many T-shirts did she sell?

We are given the equation for Emma's income:

\[
I = 100 + 12J + 5T + 7S + 5H - 8M
\]

Where:
- $ I $ is her income,
- $ J $ is the number of jeans she sells,
- $ T $ is the number of T-shirts she sells,
- $ S $ is the number of shorts she sells,
- $ H $ is the number of hats she sells, and
- $ M $ is the number of meals she buys.

We are given:
- $ I = 205 $
- $ J = 6 $
- $ S = 4 $
- $ H = 5 $
- $ M = 5 $
- We need to find $ T $.

Substituting the known values into the equation:

\[
205 = 100 + 12(6) + 5T + 7(4) + 5(5) - 8(5)
\]

Let's calculate the individual terms:

- $ 12 \times 6 = 72 $
- $ 7 \times 4 = 28 $
- $ 5 \times 5 = 25 $
- $ 8 \times 5 = 40 $

Now, substitute these values back into the equation:

\[
205 = 100 + 72 + 5T + 28 + 25 - 40
\]

Simplify the expression on the right side:

\[
205 = 185 + 5T
\]

To find $ T $, subtract 185 from both sides:

\[
205 - 185 = 5T
\]

\[
20 = 5T
\]

Finally, divide by 5:

\[
T = \frac{20}{5} = 4
\]

Emma sold 4 T-shirts that week.

Would you like more details or have any other questions?

Here are some related questions you might consider:
1. How would Emma's income change if she bought 2 fewer meals?
2. What would her income be if she sold 8 pairs of jeans instead of 6?
3. How much would she make if she didn't sell any shorts or hats?
4. What happens to her income if she sells 10 T-shirts and no other items?
5. If the price of meals increases by $2 per meal, how does that affect her income?
6. What would her income be if she sold twice as many jeans as hats?
7. How would her income change if she didn't buy any meals during the week?
8. What if Emma's base salary increased from $100 to $150 per week?

**Tip:** When working with income-related equations, breaking down the equation step by step helps ensure accuracy in solving for unknowns.

Learn how to calculate Emma's weekly income using a mathematical model that includes sales of jeans, shorts, hats, and meals expenses. Detailed step-by-step solution provided.

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