The ratio $4:3$ means for every 4 small bolts, there are 3 large bolts. If you know the total number of small bolts, think of that as the "4" in the ratio and ask: what does 1 part equal, and what do 3 parts equal?

Let $x$ be the number of large bolts this week. You know small:large is $4:3$ and you know the actual number of small bolts from Hint 1. Set up $\dfrac{4}{3} = \dfrac{\text{small}}{x}$ and solve for $x$.

In Desmos, type 12800*1.25 (or 12800*(5/4)) and look at the output. This is the number of small bolts produced this week after the 25% increase.

In a new line, type (12800*1.25/4)*3 to apply the $4:3$ ratio (divide by 4 to get 1 part, then multiply by 3 for large bolts). The value Desmos gives is the required number of large bolts this week.

Last week the factory made 12,800 small bolts.

This week, small-bolt production increases by 25%. An increase of 25% means you multiply by $1.25$ (or by $\frac{5}{4}$):

Compute this to get the number of small bolts produced this week.

The small-to-large ratio is $4:3$. This means:

• For every 4 small bolts, there are 3 large bolts.
• We can think of total production as being split into equal "parts": 4 parts small and 3 parts large.

If this week’s small-bolt total is $S$, then:

• $4$ parts $= S$ small bolts
• $1$ part $= \dfrac{S}{4}$
• $3$ parts (large bolts) $= 3 \times \dfrac{S}{4}$

Next, plug in the actual value of $S$ from Step 1.

From Step 1, you should have found that this week’s small-bolt total is $16000$.

Now use the ratio calculation from Step 2:

Compute $\frac{16000}{4} = 4000$, then multiply by 3:

So, the factory must produce $12000$ large bolts this week.