Depth · one concept from first principles · step 2 of 4

Salt balance turns a mixture into proportion

Mixing does not create or destroy salt. Write the salt in each solution, set the total equal to the salt in the c% mixture, and solve for y in terms of x.

RatiosExpressions

Your answer

How to solve

Strategy Convert to Algebra — Track the one quantity that stays the same, the grams of salt, and write each piece as percent × mass. That gives one equation, and collecting the x and y terms shows they are proportional.

1STEP 1

Salt in each solution

A p% solution of mass m g contains (p/100)m g of salt.

salt₁ = (a/100)x, salt₂ = (b/100)y

2STEP 2

Balance the salt

The mixture weighs x+y g and is c% salt, and its salt is exactly the salt from the two parts.

(a/100)x + (b/100)y = (c/100)(x+y)

3STEP 3

Clear and collect

Multiply by 100, expand, and move the x terms to one side and the y terms to the other.

ax + by = cx + cy → (a-c)x = (c-b)y

4STEP 4

Solve for y

Since b ≠ c, divide both sides by c-b.

y = ((a-c)/(c-b))x

Answer

y=((a-c)/(c-b))x

Test numbers: a=10, b=4, c=6 gives y = (4/2)x = 2x. With x=100, y=200: salt 10 + 8 = 18 g in 300 g, which is 6%. It checks.

Takeaway

In a mixture, follow the thing that does not change -- here, the salt.

Salt in each solution
Balance the salt
Clear and collect
Solve for y

Where next?

Same look, different logicsuggested

Equal xy at two points fixes the ratio

Step back and solidifysuggested

Track grams of salt through two pourings Compare two drain-outs to isolate the inflow

2 reasoning problems grow out of this type — the bridge toward competition →

A parent dashboard for the family lives at sensimlab.com.

The same shape shows up later on the AMC, as Arithmetic Expression Decomposition. Why the shapes repeat →