Paint Mixture: What Percent Yellow Must Be Added?

Mixture & Concentration 9th-10th Grade
PROBLEM
There are 4 pints of green paint in a gallon bucket made from equal amounts of yellow and blue paint. To make the mixture 80% yellow, how many pints of yellow paint must be added? Brandon claims 6 pints are needed. Explain who is correct with calculations.
There are 4 pints of green paint in a gallon bucket made from equal amounts of yellow and blue paint. To make the...

What You Will Learn

  • How to set up equations for concentration problems where adding one component changes both the numerator and denominator
  • Why you cannot simply calculate percentage differences when dealing with changing total volumes
  • The algebraic technique of cross-multiplication to solve proportions in mixture problems
  • How to verify mixture calculations by checking final percentages
  • Pattern recognition for problems where "equal amounts" means 50% of each component

Solution: Method 1 — The Concentration Equation

When we add paint to a mixture, we change both the amount of the target color and the total volume. This requires careful equation setup.

Step 1 — Analyze the current mixture

The problem states we have 4 pints of green paint made from "equal amounts" of yellow and blue. This means:

Current yellow paint = 2 pints
Current blue paint = 2 pints
Total volume = 4 pints
Current yellow percentage = 2/4 = 50%

Step 2 — Set up the target equation

Let x = pints of yellow paint to add. After adding yellow paint:

New yellow amount = 2 + x pints
New total volume = 4 + x pints
Target percentage = 80% = 0.8

The concentration equation becomes:

(Yellow paint) ÷ (Total paint) = Target percentage
(2 + x) ÷ (4 + x) = 0.8

Step 3 — Solve using cross-multiplication

Cross-multiply to eliminate the fraction:

2 + x = 0.8(4 + x)
2 + x = 3.2 + 0.8x
x - 0.8x = 3.2 - 2
0.2x = 1.2
x = 6

Step 4 — Evaluate Brandon's claim

Brandon claims 6 pints are needed. Our calculation shows x = 6, so Brandon is correct.

Solution: Method 2 — The Fixed Blue Reference

Since we're only adding yellow paint, the amount of blue stays constant at 2 pints. We can use this as our reference point.

Step 1 — Set up the percentage relationship

If the final mixture is 80% yellow, then it must be 20% blue:

Blue percentage = 100% - 80% = 20% = 0.2

Step 2 — Use the fixed blue amount

Since blue remains at 2 pints and represents 20% of the final mixture:

2 pints = 0.2 × (Total final volume)
Total final volume = 2 ÷ 0.2 = 10 pints

Step 3 — Calculate yellow paint needed

We started with 4 pints and need 10 pints total:

Yellow paint to add = 10 - 4 = 6 pints

This confirms Brandon's answer of 6 pints.

Brandon is correct. We need to add 6 pints of yellow paint to make the mixture 80% yellow.

Verification

Let's check our answer by calculating the final mixture composition:

After adding 6 pints of yellow paint:
• Yellow paint: 2 + 6 = 8 pints
• Blue paint: 2 pints (unchanged)
• Total volume: 4 + 6 = 10 pints
• Yellow percentage: 8 ÷ 10 = 0.8 = 80% ✓
• Blue percentage: 2 ÷ 10 = 0.2 = 20% ✓

The percentages add to 100% and match our target, confirming the solution is correct.

Where Students Go Wrong

✗ Calculating percentage difference directly
Some students think: "I need to go from 50% to 80%, that's a 30% increase, so I need 4 × 0.3 = 1.2 pints of yellow." This ignores that adding paint changes the total volume, making this approach completely wrong.
✗ Using the wrong total in the denominator
Students might write (2 + x) ÷ 4 = 0.8, keeping the original total of 4 pints. This gives x = 1.2 pints, which would actually make the mixture (2 + 1.2) ÷ (4 + 1.2) ≈ 60% yellow, not 80%.
✗ Mixing up the blue paint percentage
Students sometimes calculate what's needed for 80% blue instead of 80% yellow, especially when the problem mentions both colors prominently. Always double-check which color the problem is asking about.

The Pattern Behind This

Mixture concentration problems follow a standard template:

(Amount of target substance) ÷ (Total amount) = Target concentration

The key insight is that when you add pure substance, both the numerator and denominator change. If you add x units of pure substance to a mixture:

New numerator = Original target amount + x
New denominator = Original total + x

This creates the fundamental equation structure for all mixture problems where you're adding pure components to reach a target concentration.

Why This Matters

This type of calculation appears in many real-world contexts:

  • Pharmacy: Diluting or concentrating medication solutions to achieve precise dosing
  • Chemistry: Preparing solutions with specific molarity or concentration for experiments
  • Food Industry: Adjusting sugar content, alcohol percentage, or salt concentration in products
  • Manufacturing: Creating alloys with specific metal percentages or adjusting chemical compositions

What If?

1
Reverse the Target
Starting with the same 4-pint mixture that is 50% yellow and 50% blue, how many pints of blue paint must be added to make the mixture 80% blue?
Step 1 — Set up for 80% blue

Let x = pints of blue paint to add. Current: 2 pints yellow, 2 pints blue.

Step 2 — Write the equation

New blue amount = 2 + x, total volume = 4 + x. We want: (2 + x) ÷ (4 + x) = 0.8

Step 3 — Solve

2 + x = 0.8(4 + x)
2 + x = 3.2 + 0.8x
0.2x = 1.2
x = 6

Step 4 — Verify

Final: 8 pints blue, 2 pints yellow, 10 total. Blue percentage: 8/10 = 80% ✓

Answer: 6 pints of blue paint

2
Different Starting Point
You have 6 pints of green paint that is 40% yellow and 60% blue. How many pints of pure yellow must be added to make it 75% yellow?
Step 1 — Find current amounts

Current yellow: 6 × 0.4 = 2.4 pints. Current blue: 6 × 0.6 = 3.6 pints.

Step 2 — Set up equation

Let x = yellow to add. Want: (2.4 + x) ÷ (6 + x) = 0.75

Step 3 — Cross multiply and solve

2.4 + x = 0.75(6 + x)
2.4 + x = 4.5 + 0.75x
0.25x = 2.1
x = 8.4

Step 4 — Verify

Final: 10.8 pints yellow, 3.6 pints blue, 14.4 total. Yellow: 10.8/14.4 = 0.75 = 75% ✓

Answer: 8.4 pints of yellow paint

3
Dual Addition
From the original 4-pint mixture (50% yellow, 50% blue), you add both yellow and blue paint, but twice as much yellow as blue. How much of each should you add so the final mixture is 80% yellow?
Step 1 — Define variables

Let y = blue paint added, then 2y = yellow paint added (twice as much yellow).

Step 2 — Set up the equation

New yellow: 2 + 2y, New total: 4 + y + 2y = 4 + 3y. Want: (2 + 2y)/(4 + 3y) = 0.8

Step 3 — Solve

2 + 2y = 0.8(4 + 3y)
2 + 2y = 3.2 + 2.4y
-0.4y = 1.2
y = -3

Step 4 — Interpret

Since y = -3, this means we need to remove 3 pints of blue and remove 6 pints of yellow. This makes the problem impossible as stated since we can't remove more than we have.

Answer: This constraint makes 80% yellow impossible

4
Two-Stage Process
From the original 4-pint mixture, you first add 2 pints of blue paint (making it less yellow). Then you add pure yellow to bring the concentration to 80% yellow. How much yellow do you add in the second step?
Step 1 — After adding blue

Original: 2 yellow, 2 blue. Add 2 blue: now 2 yellow, 4 blue, 6 total. Yellow %: 2/6 = 33.3%

Step 2 — Set up for final target

Let x = yellow added. Want: (2 + x) ÷ (6 + x) = 0.8

Step 3 — Solve

2 + x = 0.8(6 + x)
2 + x = 4.8 + 0.8x
0.2x = 2.8
x = 14

Step 4 — Verify

Final: 16 yellow, 4 blue, 20 total. Yellow percentage: 16/20 = 0.8 = 80% ✓

Answer: 14 pints of yellow paint in the second step

Frequently Asked Questions

Set up an equation where the amount of desired color divided by the total amount equals the target percentage. In this problem, we need (2 + x) ÷ (4 + x) = 0.8, where x is the yellow paint to add. The 2 represents existing yellow, 4 + x is the new total, and 0.8 is 80%.
Adding paint changes both the numerator (amount of target color) and denominator (total volume). You can't treat it as adding 30% more yellow to go from 50% to 80% because the total volume increases. The actual calculation requires algebra to account for both changes simultaneously.
Students often multiply the current volume by the percentage increase needed. Here, they might calculate 4 × 0.3 = 1.2 pints, thinking they need 30% more to go from 50% to 80%. This ignores that adding paint increases the total volume, making the actual answer much larger.
NJ

Dr. Neven Jurkovic

Expert reviewed for mathematical accuracy and pedagogical value

NJ
Neven Jurkovic, PhD

Professor of Computer Science, Palo Alto College, Alamo Colleges District, San Antonio, TX

Developer of Algebrator

Contact

This solution was prepared with AI assistance and reviewed by Dr. Jurkovic for mathematical accuracy and pedagogical clarity.

2026-09-18

← Previous Page
Next Page →