Average — Assam Police Constable Study Guide
Among all the chapters that the SLPRB Assam Police Constable (AB & UB) paper draws from, average is the one that can be mastered fastest and used the widest. The reason is that average is not really a formula to be memorised but a way of thinking about a group of numbers. Whenever a question gives several values and asks for a single representative figure, or gives that representative figure and asks for a missing value, it is an average question in disguise. In the Assam Police test, average turns up on its own in questions about marks, wages, rainfall and ages, and it also sits quietly inside the chapters on speed, mixture and data interpretation. A candidate who learns to think in terms of the running total and the running count will find that almost every average question can be finished in a single line of arithmetic. This guide follows the approach of the Master Book: Maths Masti (D. Hurina), which always sets down the total first and treats the average as a simple division at the end.
How the Average Chapter Fits the Assam Police Test
Objective papers reward a candidate who can decide quickly and compute neatly, and average suits exactly that. The question is usually short, the numbers are usually chosen so that the answer comes out clean, and the reasoning rarely reaches beyond one multiplication or one subtraction. The chapter is also a building block. The idea of a representative value reappears in data interpretation, and the idea of a constant total reappears in speed, age and partnership problems, so time spent here is time spent on several chapters at once.
Words You Will Meet
- Mean or average: the single value obtained by sharing the total equally among all the members.
- Member: one of the values that is being averaged.
- Total: the sum that is to be shared.
- Strength: the number of members in the group.
- Deviation: the difference between a value and a chosen reference number.
- Weighted mean: a mean in which different members carry different strength.
Picture the average as an equal sharing. If the total is distributed equally among all the members, each member receives the average. This little picture makes the meaning plain and makes the formula obvious.
One Relation, Three Faces
The average equals the total divided by the strength. Turn the same sentence around, and the total equals the average multiplied by the strength, while the strength equals the total divided by the average. Every average sum is solved by filling two of the three quantities and finding the third. Two consequences are worth carrying away. If the average of n members rises by an amount, the total rises by that amount multiplied by n. And if one member is taken out and another put in and the average changes by an amount, the new member differs from the old by that amount multiplied by n. These two consequences turn many questions into mental arithmetic.
Handling a Straight Run of Numbers
When the values are equally spaced, the average lies exactly at the centre of the run and nothing needs to be added. For an odd number of values the centre is one of the values; for an even number the centre falls between the two middle values. A run of consecutive whole numbers beginning at one has an average of half of one more than the strength. A run of consecutive even numbers has an average of one more than the strength, and a run of consecutive odd numbers has an average equal to the strength. Roll numbers, page numbers and consecutive years are all handled at once with this idea.
The Shift or Short Method for Big Values
Long columns of large numbers are tiring to add and easy to mis-add. Instead, choose a convenient round number near the values as a reference, note how far each value lies from it, average those small differences, and put the reference back. Because the differences are small and often cancel each other, the work becomes much shorter. Reach for this method when a table of figures is given in a data interpretation set, or when the numbers share their leading digits.
Blending Groups of Unequal Size
When several groups are merged, the average of the whole must be built from the totals, never from the group averages. Suppose a group of fifteen members has an average of forty and a group of five members has an average of eighty. The first contributes six hundred and the second four hundred, so the merged total is one thousand over twenty members, giving an average of fifty. Taking the mid-value of forty and eighty would give sixty, which is wrong. The mid-value of two averages is correct only in the special case where the group strengths are equal; in every other case the totals must be worked out first.
When One Member Joins, Leaves or Is Swapped
To see the effect of a change, compare the total before and after. If the total rises by a known amount and the strength rises by one, the new member must be the old average plus that amount, and reversing the argument gives a member who leaves. In a swap, the incoming member is greater than the outgoing member by the change in the average multiplied by the unchanged strength. These questions look wordy but are really one comparison of totals, and the candidate who writes both totals down avoids the long method entirely.
Questions Set in Speed, Age and Money
Average speed is a question about totals: the whole journey's distance shared over the whole journey's time. It is not the average of the various speeds, unless those speeds were used for equal times. For two equal halves of a journey covered at different speeds, the average speed takes the harmonic form, twice the product of the two speeds divided by their sum. Average age questions are ordinary average questions in a familiar costume: the total of the ages changes as members join or leave, and the average follows the total. Money questions about average income, average wage or average expenditure behave in exactly the same way.
Quick Methods for the Exam Hall
Begin by writing the total and the strength, even when they are not asked for, because they organise the whole question. For equally spaced values, jump straight to the centre. Use a reference number when the values are large and crowded. For a swap, use the change in the total rather than the whole average. If the groups differ in strength, resist the temptation to average the averages. Finally, check the result, because an average must always lie between the smallest and the largest value of the group.
Where Candidates Lose Marks
- Combining group averages directly when the groups differ in strength.
- Forgetting that the strength changes when a member is added or removed.
- Treating average speed as the mean of the speeds of the journey.
- Expecting the average to be a whole number and then rounding it wrongly.
- Losing track of the total after a member joins or leaves.
- Neglecting to change the strength when the group grows or shrinks.
Weekly Practice Routine
- Revise the one relation and its two rearrangements.
- Practise runs of equally spaced numbers.
- Practise the reference-number method on long lists.
- Work on merged groups of unequal strength.
- Finish with speed, age and swap questions, and revise the totals each week.
Common Questions Answered
Can the average be larger than every value in the group?
No. The average always lies between the smallest and the largest value of the group, and it can equal one of them only when all the values are equal. This is a quick way to reject an impossible option.
What happens to the average if every value is raised by the same number?
The average is raised by exactly that number, because the total rises by that number multiplied by the strength and the strength itself does not change. The same reasoning works if every value is reduced by the same number.
How do I choose between adding the totals and using a reference number?
If the values are small or few, add the totals directly. If the values are large, close together or given in a long table, use a convenient reference number and average the deviations, which keeps the arithmetic short.
Why does removing a small value usually raise the average?
Because the removed value was pulling the average down. Once it is gone, the same total is shared among fewer members, so the remaining average is higher. Writing the two totals side by side makes this clear at a glance.
What is the average of a run of consecutive odd numbers?
For a run of consecutive odd numbers beginning at one, the average equals the number of terms. The first five odd numbers, one, three, five, seven and nine, add to twenty-five, and dividing by five gives an average of five.