Rank from the top and the bottom

Learning outcome

You can turn a rank from the top and a rank from the bottom into the size of the class.

Concepts and reasons

A class rank is the same count as a place in a row. The top is one end. The bottom is the other end. The student is counted in both ranks, so class size = rank from the top + rank from the bottom - 1. A student who is 5th from the top and 16th from the bottom sits in a class of 5 + 16 - 1 = 20. When the class size is known, the other rank is class size minus this rank, plus 1. In a class of 40, rank 9 from the top is rank 40 - 9 + 1 = 32 from the bottom. People who scored higher than a student are the ranks above that student. Rank 6 means 5 people scored higher, if no two students share a rank. This lesson uses unique ranks only. If two students have the same marks, the question has to say how ties are counted. Do not invent that count.

Worked examples

1. 5th from the top and 16th from the bottom. Class size = 5 + 16 - 1 = 20.
2. A class has 40 students. Rank 9 from the top is rank 32 from the bottom, because 40 - 9 + 1 = 32.
3. A student is ranked 6th, and no two students share a rank. Students who scored higher = 6 - 1 = 5.

Common traps

5 + 16 = 21 counts the student twice. The class has 20. Rank 6 does not mean 6 people scored higher. The rank itself is that student. A shared mark is outside this rule. Wait until the question says whether those two take one rank or two.

Practice questions

1. A student is 8th from the top and 13th from the bottom. How many students are in the class?
2. A class has 25 students. What is the rank from the bottom of the student who is 4th from the top?
3. A student is ranked 10th, and no two students share a rank. How many scored higher?

Answers and explanations

1. 20. 8 + 13 - 1 = 20.
2. 22. 25 - 4 + 1 = 22.
3. 9. The 10th rank is that student, so 9 ranks are above.

Analogy

The mark list is a row stood on its end. Top and bottom are left and right. The same minus 1 stops you counting the student twice.

Quick reference

• Class size = from the top + from the bottom - 1.
• Other end = class size - this rank + 1.
• People above a unique rank = rank - 1.
• A tie is not in this rule.