A Simple Solution Using a Balance Diagram
A balance diagram is a convenient method for quickly finding the amounts and concentrations when mixing, using a diagram like a seesaw.
It is especially useful for problems in which you need to find the amounts to mix, such as “I want to mix ○ g of each.”
What You Will Learn
How to use a balance diagram:
Let the concentration on the left be a%, the concentration on the right be b%, and the concentration after mixing be c%.
In this case,
Weight of A : Weight of B ＝（b － c）：（c － a）
※It is easier to remember this by noting that the “length of the arm” and the “weight” of a balance correspond in reverse.
Example
We want to mix 3% salt water and 12% salt water to make 450 g of 8% salt water. How many grams of each should be mixed?
Solution (balance diagram):
3% 8% 12%
|←5→|←4→|（Difference on the left: 8－3＝5; difference on the right: 12－8＝4）
① Match the differences in reverse to find the ratio of the weights:
Amount of 3% salt water : Amount of 12% salt water ＝ 4 : 5
② The sum of the ratios is 4＋5 ＝ 9, so:
3% salt water: 450 × 4/9 ＝ 200g
12% salt water: 450 × 5/9 ＝ 250g
We want to mix 3% salt water and 12% salt water to make 450 g of 8% salt water. How many grams of each should be mixed?
Solution (balance diagram):
3% 8% 12%
|←5→|←4→|（Difference on the left: 8－3＝5; difference on the right: 12－8＝4）
① Match the differences in reverse to find the ratio of the weights:
Amount of 3% salt water : Amount of 12% salt water ＝ 4 : 5
② The sum of the ratios is 4＋5 ＝ 9, so:
3% salt water: 450 × 4/9 ＝ 200g
12% salt water: 450 × 5/9 ＝ 250g
Key Points
Remember
- In a balance diagram, the difference in concentration and the weight of the salt water correspond in reverse (they are inversely proportional).
- The concentration c after mixing must always be between a and b.
- Balance diagrams can be used for various problems, such as work-rate problems, in addition to salt water concentration problems.
Practice Problems
- Mix 5% and 15% salt water to make 400 g of 9% salt water. How many grams of each are needed?
- You have 300 g of 7% salt water and want to make it 10% concentration. How many grams of salt should be added?
- If 2% and 8% salt water are mixed in a ratio of 3:2, what will the concentration be?
Answers and Explanations
- Answer: 5% salt water = 240g; 15% salt water = 160g
Explanation: Use a balance diagram to mix 5% and 15% salt water to make 9% salt water. The arm lengths are (9−5)=4 and (15−9)=6. Since the ratio is reversed, the amount of 5% salt water : amount of 15% salt water = 6:4 = 3:2. Divide the total of 400g in the ratio 3:2 → 5% salt water=400×3/5=240g, 15% salt water=400×2/5=160g. - Answer: 10g of salt should be added
Explanation: Let the amount of salt added be xg. The original amount of salt = 300×0.07=21g. (21+x)÷(300+x)=0.10 → 21+x=0.10×(300+x) → 21+x=30+0.10x → 0.90x=9 → x=10g. Therefore, add 10g of salt. (Check: (21+10)÷(300+10)×100=31÷310×100=10% ✓) - Answer: 4.4%
Explanation: Mix 2% and 8% salt water in a ratio of 3:2. Let the total weight be 5, with the amounts assumed to be 3 and 2, respectively. The concentration is (0.02×3＋0.08×2)÷(3＋2)×100＝0.22÷5×100＝4.4%.