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Mixture &
Comparison Problems

Learn to solve problems where different quantities combine or need to be compared using equations: from mixing solutions to finding unknown amounts.

SSAT Middle Level: Quantitative

What Are Mixture Problems?

A mixture problem asks you to combine two or more things: different priced items, solutions with different concentrations, or groups with different properties: and find a resulting total value, quantity, or average.

Coins
A jar has nickels and dimes. How much money is there in total?
Trail Mix
Mixing nuts at \(\$8\)/lb with raisins at \(\$3\)/lb. What is the combined cost?
Solutions
Blending 20% juice with 80% juice. What percent is the final mix?

What Are Comparison Problems?

A comparison problem gives you a relationship between two or more unknown quantities: one is more than, less than, or a multiple of another: and asks you to find each quantity.

Why These Problems Matter

Mixture and comparison problems appear on almost every SSAT Quantitative section. They test your ability to set up and solve equations from real-world scenarios.

1
Real-life skills: budgeting, cooking recipes, and shopping all involve mixing and comparing amounts.
2
Equation fluency: these problems strengthen your ability to translate English into algebra.
3
Test frequency: 2 to 4 questions per SSAT section typically involve mixture or comparison setups.

The Total Value Equation

Every mixture problem follows one master formula. When you combine items, the total value equals the sum of each part's value:

Item A
\(\text{qty} \times \text{price}\)
 + Item B
\(\text{qty} \times \text{price}\)
 = Total Value Each part contributes to the whole

Setting Up Comparison Equations

For comparison problems, you write two equations: one for the relationship between the quantities and one for the total.

\(x\)
\(3x\)
\(= 48\)
 Tom Maria
\(x + 3x = 48 \rightarrow 4x = 48 \rightarrow x = 12\)

Mixture vs Comparison

Mixture Comparison
What you combine Items with different rates/prices Unknowns with a given relationship
Key equation \(\text{qty}_1 \times \text{rate}_1 + \text{qty}_2 \times \text{rate}_2 = \text{total}\) \(x + (\text{multiplier} \cdot x) = \text{total}\)
Typical clue words "mix", "blend", "combined cost" "times as many", "more than", "fewer than"
Number of unknowns Usually 1 (other is derived) Usually 2 (related by a rule)
Example Coins: nickels + dimes Ages: twice as old

How to Solve Step by Step

1
Read carefully: identify what is being mixed or compared and what the question asks.
2
Define a variable: let \(x\) represent the unknown quantity (usually the smaller or simpler one).
3
Write expressions for all other quantities in terms of \(x\).
4
Set up the equation: use the total value, total quantity, or given relationship.
5
Solve and check: solve for \(x\), find other quantities, and verify the answer makes sense.

Example 1: Coin Mixture

1
Let \(x\) = number of nickels. Then dimes = \(24 - x\).
2
Value equation: \(0.05x + 0.10(24 - x) = 1.65\)
3
Distribute: \(0.05x + 2.40 - 0.10x = 1.65\)
4
Combine: \(-0.05x = -0.75 \rightarrow x = 15\)
5
Answer: 15 nickels and 9 dimes. Check: \(15 \times 0.05 + 9 \times 0.10 = 0.75 + 0.90 = 1.65\) ✓

Example 2: Age Comparison

1
Let \(x\) = Alex's age. Then Sam's age = \(x + 5\).
2
Total equation: \(x + (x + 5) = 31\)
3
Simplify: \(2x + 5 = 31 \rightarrow 2x = 26 \rightarrow x = 13\)
4
Answer: Alex is 13, Sam is 18. Check: \(13 + 18 = 31\) ✓

Example 3: Ticket Prices

1
Let \(a\) = adult tickets. Then child tickets = \(200 - a\).
2
Value equation: \(12a + 7(200 - a) = 1950\)
3
Distribute: \(12a + 1400 - 7a = 1950 \rightarrow 5a = 550 \rightarrow a = 110\)
4
Answer: 110 adult and 90 child tickets. Check: \(110 \times 12 + 90 \times 7 = 1320 + 630 = 1950\) ✓

Example 4: Percent Mixture

6 liters 20% juice = 1.2 L juice + 4 liters 80% juice = 3.2 L juice = 10 liters 4.4 L juice = 44%
\(\text{Total juice} \div \text{Total volume} = 4.4 \div 10 = 44\%\)

Translating Words to Equations

English Phrase Math Translation
"3 times as many" \(3x\)
"5 more than" \(x + 5\)
"7 fewer than" \(x - 7\)
"twice as much" \(2x\)
"the sum is 40" \(x + y = 40\)
"total cost is $60" \(\text{qty}_1 \times \text{price}_1 + \text{qty}_2 \times \text{price}_2 = 60\)

Common Mistakes to Avoid

✕ Forgetting Units
If nickels are worth \(\$0.05\), not \(\$5\). Always convert to the same unit (cents or dollars) before writing the equation.
✕ Wrong Variable Assignment
If the total is 24 coins and \(x\) = nickels, then dimes = \(24 - x\), not another separate variable with no link.
✕ Averaging Percents Directly
You cannot just average 20% and 80% to get 50%. You must weight by quantity: \(\frac{0.20 \times 6 + 0.80 \times 4}{10} = 44\%\).

Summary: Your Toolkit

Define Variables
Pick a letter for the unknown. Express everything else in terms of it.
Build Equations
Use the total value formula for mixtures or the relationship equation for comparisons.
Solve & Simplify
Combine like terms, isolate \(x\), and compute the answer.
Check Your Work
Plug your answer back into the original problem to confirm it makes sense.

Question 1 of 5

A store sells apples for \(\$2\) each and oranges for \(\$3\) each. A customer buys 10 fruits for a total of \(\$24\). How many apples did they buy?

Question 2 of 5

Lisa has 4 times as many books as Mike. Together they have 35 books. How many books does Lisa have?

Question 3 of 5

A piggy bank has quarters and dimes. There are 18 coins worth \(\$2.85\) total. How many quarters are there?

Question 4 of 5

You mix 10 pounds of cashews at \(\$9\) per pound with some peanuts at \(\$4\) per pound. The mixture costs \(\$6\) per pound. How many pounds of peanuts did you add?

Question 5 of 5

A number is 12 less than three times another number. Their sum is 56. What is the larger number?