Arithmetic

Mean, Median & Mode

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The average weight of students in a class increases by $600$ gm when some new students join the class. If the average weight of the new students is $3$ kg more than the average weight of the original students, then the ratio of the number of original students to the number of new students is

$1:4$

$1:2$

$4:1$

$3:1$

✅ Correct Option: 3

We need to find how the original and new students are distributed to create this change in average.

Let us define what we know:

Original students have some average weight (let's call it A kg)

New students join with average weight = A + 3 kg

After new students join, the overall average increases by 0.6 kg

So the final average = A + 0.6 kg

When groups with different averages combine, we can use the alligation method to find their ratio. This method works by looking at how far each group's average is from the final combined average.

Final average = A + 0.6 kg

Original students' average = A kg

Deviation of original students = A - (A + 0.6) = -0.6 kg (they pull the average down)

New students' average = A + 3 kg

Deviation of new students = (A + 3) - (A + 0.6) = 2.4 kg (they pull the average up)

For the averages to balance out:

(Number of original students) × (their deviation) = (Number of new students) × (their deviation)

Since deviations are in opposite directions, we take absolute values:

(Number of original students) × 0.6 = (Number of new students) × 2.4

Ratio of original to new students = 2.4 : 0.6 = 4 : 1

The alligation method works because in a weighted average, the groups that are farther from the final average need fewer members to balance those closer to it. Here, since new students have a much higher average (2.4 kg above final), we need fewer of them to balance the original students.

The reference solution uses a smart shortcut by setting the original average as 0. This doesn't change the ratios but makes calculations cleaner:

Original average = 0

New average = 3

Final average = 0.6

Deviations: 0.6 and 2.4

Ratio = 2.4 : 0.6 = 4 : 1

Answer: 4 : 1

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The average weight of students in a class increases by $600$ gm when some new students join the class. If the average weight of the new students is $3$ kg more than the average weight of the original students, then the ratio of the number of original students to the number of new students is