A youtuber decides to launch limited edition merchandise that includes bags, shoes and jackets. In 2018, she launches the merchandise in the ratio 2:3:4, respectively. In 2019, she removes 'n' items from each type of merchandize launched in 2018 so that the new ratio is 3:5:7.

Similarly, in 2020, she removes 10 items from each type of merchandise produced in 2019 so that the ratio is 1:2:3.

What is the total number of items that were removed from the merchandise launched in 2019?

Let's break down the information given step by step.

### Step 1: Merchandise in 2018
In 2018, the merchandise is launched in the ratio \( 2:3:4 \) for bags, shoes, and jackets respectively. Let the actual quantities of bags, shoes, and jackets launched in 2018 be:

\[
2x, \ 3x, \ 4x
\]
where \( x \) is a constant multiplier representing the number of sets produced in that year.

### Step 2: Merchandise in 2019
In 2019, \( n \) items are removed from each type of merchandise, leading to new quantities for bags, shoes, and jackets:

- Bags: \( 2x - n \)
- Shoes: \( 3x - n \)
- Jackets: \( 4x - n \)

The new ratio becomes \( 3:5:7 \). This gives the following system of equations:

\[
\frac{2x - n}{3} = \frac{3x - n}{5} = \frac{4x - n}{7}
\]

### Step 3: Merchandise in 2020
In 2020, another 10 items are removed from each type of merchandise. So, the quantities after this removal become:

- Bags: \( 2x - n - 10 \)
- Shoes: \( 3x - n - 10 \)
- Jackets: \( 4x - n - 10 \)

Now, the ratio is \( 1:2:3 \), giving another system of equations:

\[
\frac{2x - n - 10}{1} = \frac{3x - n - 10}{2} = \frac{4x - n - 10}{3}
\]

### Step 4: Solving the System of Equations

Let's first solve for \( n \) and \( x \) using the equations from 2019 and 2020.

#### Equation from 2020:
\[
\frac{2x - n - 10}{1} = \frac{3x - n - 10}{2}
\]

Cross-multiplying:
\[
2(2x - n - 10) = 3x - n - 10
\]
Expanding both sides:
\[
4x - 2n - 20 = 3x - n - 10
\]
Simplifying:
\[
4x - 3x = 2n - n + 10 - 20
\]
\[
x = n - 10
\]
Thus, \( n = x + 10 \).

#### Equation from 2019:
Now use the fact that \( n = x + 10 \) in the system from 2019.

Take the first two ratios:
\[
\frac{2x - n}{3} = \frac{3x - n}{5}
\]
Cross-multiplying:
\[
5(2x - n) = 3(3x - n)
\]
Expanding:
\[
10x - 5n = 9x - 3n
\]
Simplifying:
\[
10x - 9x = 5n - 3n
\]
\[
x = 2n
\]
Substitute \( n = x + 10 \) into \( x = 2n \):
\[
x = 2(x + 10)
\]
Expanding:
\[
x = 2x + 20
\]
Simplifying:
\[
x - 2x = 20
\]
\[
-x = 20
\]
\[
x = -20
\]

Thus, \( x = 20 \). Now substitute \( x = 20 \) into \( n = x + 10 \):
\[
n = 20 + 10 = 30
\]

### Step 5: Total Number of Items Removed
In 2019, \( n = 30 \) items were removed from each type of merchandise. There are three types of merchandise (bags, shoes, jackets), so the total number of items removed is:

\[
3 \times 30 = 90
\]

Thus, the total number of items removed from the merchandise launched in 2019 is **90**.

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. How would the result change if the ratios were reversed?
2. What if only a certain percentage of items were removed instead of a fixed number?
3. Can you derive a formula for \( n \) if the ratios vary by more than one factor?
4. How would this affect the profit margin if the prices of the merchandise were known?
5. What happens if a fourth category of merchandise is introduced with a new ratio?

**Tip**: Always double-check that the ratios are correctly applied in equations to avoid calculation errors.

A YouTuber decides to launch limited edition merchandise that includes bags, shoes, and jackets. In 2018, she launches the merchandise in the ratio 2:3:4, respectively. In 2019, she removes 'n' items from each type of merchandise launched in 2018 so that the new ratio is 3:5:7. Similarly, in 2020, she removes 10 items from each type of merchandise produced in 2019 so that the ratio is 1:2:3. What is the total number of items that were removed from the merchandise launched in 2019?

Learn how to solve a ratio and proportion problem involving a YouTuber's limited edition merchandise launch with bags, shoes, and jackets. This step-by-step guide covers algebra, systems of equations, and ratio calculations. Suitable for students in Grades 8-10.

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