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GMAT Quant
Mixture Equations
Solve mixture problems by conserving total amount and the amount of pure ingredient.
Core idea
The pure ingredient is what you track.
In a mixture, pure solute = concentration x total amount. When containers are mixed, the pure solute amounts add.
A 25% solution
- Total solution
- 8 L
- Concentration
- 0.25
- Pure ingredient
- 0.25*8 = 2 L
- Other material
- 6 L
Two-equation method
For exact amounts, write two equations.
Equation 1 tracks total volume. Equation 2 tracks pure ingredient.
General setup
- Total amount
- x + y = T
- Pure ingredient
- c1*x + c2*y = ct*T
- Unknowns
- x and y
Setup
Example: make 12 L of 35% from 20% and 50%.
Let x be liters of 20% and y liters of 50%. Then x + y = 12 and 0.20x + 0.50y = 4.2.
Solve the example.
- From total
- x = 12 - y
- Substitute
- 0.20(12-y)+0.50y=4.2
- y
- 6 L of 50%
- x
- 6 L of 20%
Check
Use the target concentration as a sanity check.
A target closer to the low concentration should require more low-concentration solution; closer to the high should require more high.
Mini application
Try this: make 10 L of 28% from 20% and 40%.
Set up the two equations before scrolling.
Solution
Use 6 L of 20% and 4 L of 40%.
x + y = 10 and 0.20x + 0.40y = 2.8. Solving gives y = 4 and x = 6.
Remember this
Conserve total amount and pure ingredient.
Those two quantities give the reliable setup for exact mixture amounts.