Solve the system: X * 3.80 + Y * 4.20 = 6627.75 and X + Y = 1628.

Step-by-step solution

1. 1

   Compute the average price per unit. Dividing the total value by the total count gives the blended rate:

   $\bar p=\frac{6627.75}{1628}=4.071099\ldots$

   It lies between $3.80$ and $4.20$, as it must — an average outside that range would signal an inconsistent problem.

2. 2

   Use the alligation (lever) relation. The distances from the average to the two prices are inversely proportional to the quantities:

   $\frac{X}{Y}=\frac{4.20-\bar p}{\bar p-3.80}=\frac{0.128901}{0.271099}$

   This is the weighted-average shortcut: the cheaper item pulls the mean down in proportion to how much of it there is.

3. 3

   Convert to a single equation and solve exactly. Rather than working with the rounded average, clear the decimals directly. Multiplying the value equation by $100$ and substituting $X=1628-Y$:

   $380(1628-Y)+420Y=662775\;\Longrightarrow\;618640+40Y=662775$

   $40Y=44135\;\Longrightarrow\;Y=1103.375$

4. 4

   Find $X$ from the total.

   $X=1628-1103.375=524.625$

   Both values terminate exactly in decimals here, since $44135/40$ has denominator $8$ after reduction.

5. 5

   Check the value equation and the ratio. $3.80(524.625)+4.20(1103.375)=1993.575+4634.175=6627.75\;\checkmark$. The ratio check: $\tfrac{524.625}{1103.375}=0.4755$, and $\tfrac{0.128901}{0.271099}=0.4755\;\checkmark$, confirming the alligation reasoning agrees with the algebra.