Mixing Paint to a Target Ratio
A contractor mixes tinted paint by combining blue concentrate and white base in a fixed ratio of 3 parts blue to 8 parts white by volume. One job requires 66 liters of the finished mixture. The contractor has 20 liters of blue concentrate on hand and an effectively unlimited supply of white base. A helper proposes simply adding 3 liters of blue to 8 liters of white per batch and repeating until 66 liters is reached, but 11-liter batches do not divide evenly into 66 liters in a way that keeps the ratio clean, and the contractor wants the exact amount of each ingredient for the whole 66 liters in a single mix, plus a check that the blue on hand is sufficient.
1. How many liters of blue concentrate and white base are needed for exactly 66 liters of the 3:8 mixture?
Read the worked answer
Total ratio parts: 3 + 8 = 11 parts.
Each part = 66 ÷ 11 = 6 liters.
Blue = 3 × 6 = 18 liters. White = 8 × 6 = 48 liters.
Check: 18 + 48 = 66 liters, and 18:48 reduces to 3:8 (divide both by 6). Correct.
2. Is the 20 liters of blue on hand enough, and how much blue would remain after the mix?
Read the worked answer
Needed blue = 18 liters; on hand = 20 liters. Since 20 > 18, there is enough.
Remaining blue = 20 − 18 = 2 liters.
The 2-liter surplus is not enough to matter for this job but could seed a smaller batch: at 3:8, 2 liters of blue would pair with (8/3) × 2 ≈ 5.33 liters of white to make about 7.33 liters - useful to note but not required here.
3. The helper suggests repeating 3-blue-to-8-white batches. If the contractor instead made 5 such 11-liter batches, how far short of 66 liters would that be, and what would the ratio problem be?
Read the worked answer
5 batches × 11 liters = 55 liters, which is 66 − 55 = 11 liters short - exactly one more batch.
So 5 batches under-fill the job; 6 batches would give 66 liters and would actually preserve the ratio (6 × 3 = 18 blue, 6 × 8 = 48 white), matching the single-mix answer. The helper's method is workable only if the batch count is a whole number that lands on 66; here 6 batches works, but the single-mix calculation is faster and avoids rounding errors if the target volume were not a multiple of 11.