4. Weighted averages and mixtures
Number line method: distances from the mean are in INVERSE ratio to the group sizes.
Core ideas
- A weighted average always sits between the two group values, closer to the bigger group.
- Number line method: distances from each group value to the mixed average are in INVERSE ratio to the group sizes.
- weight1/weight2 = (avg2 - mix)/(mix - avg1): the far group is the small group.
- Works for any mixture: solutions, prices, test scores, speeds over equal times.
Worked example 1
A class of men and women averages 168 cm tall. Men average 174 cm and women average 160 cm. What fraction of the class is men?
Show solution
Put 160, 168, 174 on a number line. Distance from women's average to the mix is 168 - 160 = 8; distance from men's average is 174 - 168 = 6. Group sizes are in inverse ratio of distances, so men : women = 8 : 6 = 4 : 3. Men are 4/7 of the class. Check: (4*174 + 3*160)/7 = (696 + 480)/7 = 1176/7 = 168.
Worked example 2
How many liters of a 30% acid solution must be mixed with 6 liters of an 80% acid solution to get a 50% solution?
Show solution
Distances to the 50% target: the 30% solution is 20 points away and the 80% solution is 30 points away. Volumes are in inverse ratio: (30% solution) : (80% solution) = 30 : 20 = 3 : 2. Since the 80% part is 6 liters (the 2 in the ratio, scaled by 3), the 30% part is 9 liters. Check: acid = 0.3*9 + 0.8*6 = 2.7 + 4.8 = 7.5 liters out of 15 total, which is 50%.
Practice set
Question 1
Group A has 10 people with an average weight of 50 kg. When Group A is combined with Group B, which has 20 people, the average weight of all 30 people is 62 kg. What is the average weight of Group B?
Work with totals: combined total minus Group A's total gives Group B's total.
Combined total = 30*62 = 1860. Group A total = 10*50 = 500. Group B total = 1860 - 500 = 1360, so Group B average = 1360/20 = 68 kg. Choice E (74) comes from assuming the distances from 62 are equal rather than in inverse ratio to group sizes.
Question 2
In a company, the average age of the men is 30 and the average age of the women is 25. If the average age of all employees is 27, what percent of the employees are women?
Group sizes are in inverse ratio to each group's distance from the overall average.
Distance from 27: men are 3 away, women are 2 away. Sizes are in inverse ratio, so men:women = 2:3. Women are 3/5 of the employees = 60 percent. Choice A (40) is the men's share, the classic ratio flip.
Question 3
In a class, 20 boys have an average score of 70 and 30 girls have an average score of 80. What is the average score of the whole class?
The overall average sits closer to the larger group's average.
Total points = 20*70 + 30*80 = 1400 + 2400 = 3800. Average = 3800/50 = 76. Choice B (75) is the trap of averaging 70 and 80 without weighting.
Question 4
A 40 percent acid solution is mixed with a 70 percent acid solution to produce 18 liters of a 50 percent acid solution. How many liters of the 70 percent solution are used?
The distances from 50 on the number line (10 and 20) give the ratio of the two amounts, inverted.
Distances from 50: 50 - 40 = 10 and 70 - 50 = 20. Amounts are in inverse ratio, so 40 percent solution : 70 percent solution = 20:10 = 2:1. The 70 percent solution is 1/3 of 18 liters = 6 liters. Choice E (12) is the amount of the 40 percent solution.