Weighted Average and Alligation (Seesaw Analogy)

Practice flashcards on weighted averages and alligation, including the seesaw analogy and example problems.

What is weighted average?

An average where each quantity is weighted by its amount; total value divided by total quantity.

How is the weighted average cost price calculated?

Sum of (quantityi × pricei) for all items divided by the total quantity.

In the two-quantity example, 68 kg at 63 and 85 kg at 81, what is the average cost price?

73 rupees per kg.

What is the seesaw principle for weighted averages?

The heavier weight sits closer to the pivot; distance is inversely proportional to weight; balance requires w1×d1 = w2×d2.

What does w1×d1 = w2×d2 represent?

The balance condition on a seesaw; the product of weight and distance from the pivot is equal on both sides.

What does 'distance from the mean' mean in alligation?

Distances from the mean of the two quantities are used to allocate proportions; they reflect how far each quantity is from the overall mean.

How is the distance ratio determined in alligation?

Distance ratio is the inverse of the weight ratio: if weights are w1:w2, distances are w2:w1.

Where does the combined mean lie relative to the two group means?

The overall mean lies between the two means.

What is the two-group mean formula?

M = (N1×M1 + N2×M2) / (N1 + N2).

Example: combined mean of 45 people at 173 cm and 75 people at 197 cm?

188 cm.

Example: price problem with bananas to gain 35%?

Selling price per kg is 20.70 rupees.

Example: two groups of students with means 438 and 487?

Overall mean is 473.

Key principle of alligation?

Distances from the mean are divided in the ratio of the opposite weight; weight and distance are inversely related.

What is weighted average?

An average where each quantity is weighted by its amount; total value divided by total quantity.

A simple average gives equal importance to all data points, summing them and dividing by the count. A weighted average assigns varying importance (weights) to each data point before summing, then divides by the sum of the weights, reflecting their relative contribution.

How does a simple average differ from a weighted average?

How is the weighted average cost price calculated?

Sum of (quantityi × pricei) for all items divided by the total quantity.

quantityi represents the amount or frequency of the i^{th} item or group, and pricei (or value_i) represents the value or cost associated with that i^{th} amount.

In the weighted average cost price formula, \sum (quantityi \times pricei) / \sum quantityi , what do quantityi and price_i represent?

In the two-quantity example, 68 kg at 63 and 85 kg at 81, what is the average cost price?

73 rupees per kg.

What is the seesaw principle for weighted averages?

The heavier weight sits closer to the pivot; distance is inversely proportional to weight; balance requires w1×d1 = w2×d2.

What does w1×d1 = w2×d2 represent?

The balance condition on a seesaw; the product of weight and distance from the pivot is equal on both sides.

What does 'distance from the mean' mean in alligation?

Distances from the mean of the two quantities are used to allocate proportions; they reflect how far each quantity is from the overall mean.

How is the distance ratio determined in alligation?

Distance ratio is the inverse of the weight ratio: if weights are w1:w2, distances are w2:w1.

The alligation method is especially useful for problems involving mixtures, where two different components (e.g., liquids of different concentrations, items of different prices) are combined to achieve a desired average property. It helps determine the ratio in which the components should be mixed.

What types of problems is the alligation method particularly useful for solving?

Where does the combined mean lie relative to the two group means?

The overall mean lies between the two means.

What is the two-group mean formula?

M = (N1×M1 + N2×M2) / (N1 + N2).

Example: combined mean of 45 people at 173 cm and 75 people at 197 cm?

188 cm.

Example: price problem with bananas to gain 35%?

Selling price per kg is 20.70 rupees.

Example: two groups of students with means 438 and 487?

Overall mean is 473.

Key principle of alligation?

Distances from the mean are divided in the ratio of the opposite weight; weight and distance are inversely related.

What is weighted average?

An average where each quantity is weighted by its amount; total value divided by total quantity.

How does a simple average differ from a weighted average?

A simple average gives equal importance to all data points, summing them and dividing by the count. A weighted average assigns varying importance (weights) to each data point before summing, then divides by the sum of the weights, reflecting their relative contribution.

How is the weighted average cost price calculated?

Sum of (quantityi × pricei) for all items divided by the total quantity.

In the weighted average cost price formula, \sum (quantityi \times pricei) / \sum quantityi , what do quantityi and price_i represent?

quantityi represents the amount or frequency of the i^{th} item or group, and pricei (or value_i) represents the value or cost associated with that i^{th} amount.

In the two-quantity example, 68 kg at 63 and 85 kg at 81, what is the average cost price?

73 rupees per kg.

What is the seesaw principle for weighted averages?

The heavier weight sits closer to the pivot; distance is inversely proportional to weight; balance requires w1 \times d1 = w2 \times d2.

What does w1 \times d1 = w2 \times d2 represent?

The balance condition on a seesaw; the product of weight and distance from the pivot is equal on both sides.

What does 'distance from the mean' mean in alligation?

Distances from the mean of the two quantities are used to allocate proportions; they reflect how far each quantity is from the overall mean.

How is the distance ratio determined in alligation?

Distance ratio is the inverse of the weight ratio: if weights are w1:w2, distances are w2:w1.

What types of problems is the alligation method particularly useful for solving?

The alligation method is especially useful for problems involving mixtures, where two different components (e.g., liquids of different concentrations, items of different prices) are combined to achieve a desired average property. It helps determine the ratio in which the components should be mixed.

Where does the combined mean lie relative to the two group means?

The overall mean lies between the two means.

What is the two-group mean formula?

M = (N1 \times M1 + N2 \times M2) / (N1 + N2).

Example: combined mean of 45 people at 173 cm and 75 people at 197 cm?

188 cm.

Example: price problem with bananas to gain 35%?

Selling price per kg is 20.70 rupees.

Example: two groups of students with means 438 and 487?

Overall mean is 473.

Key principle of alligation?

Distances from the mean are divided in the ratio of the opposite weight; weight and distance are inversely related.

In the seesaw principle w1 \times d1 = w2 \times d2 what do w1, w2, d1, d2 represent in an alligation problem?

w1 and w2 are the quantities (weights) of the two components, and d1 and d2 are their respective distances from the combined mean.

According to the alligation method's inverse relationship, if ComponentA has a larger 'weight' (quantity) than ComponentB, how will its distance from the combined mean compare to Component_B's distance?

ComponentA's distance from the combined mean will be proportionally smaller than ComponentB's, as distance is inversely proportional to weight.

Given two groups with means M1 and M2, how does the combined mean relate to these individual means?

The combined mean M will always lie between M1 and M2.