Position from the left and the right

Learning outcome

You can find how many people are in a row when one person is counted from both ends.

Concepts and reasons

Count the same person once. A place from the left and a place from the right both include that person, so the total is left place plus right place, minus 1. If a person is 6th from the left and 9th from the right, the row has 6 + 9 - 1 = 14 people. The minus 1 is the whole rule. Adding the two places counts that person twice. When the total is already known, the other end is total minus this place, plus 1. In a row of 20, the 7th person from the left is 20 - 7 + 1 = 14th from the right. This rule is for one row in a line, with each person in one place. It is not for a circle.

Worked examples

1. A person is 6th from the left and 9th from the right. Total = 6 + 9 - 1 = 14.
2. A row has 20 people. The person who is 7th from the left is 14th from the right, because 20 - 7 + 1 = 14.
3. A person is 4th from the left and 4th from the right. Total = 4 + 4 - 1 = 7.

Common traps

6 + 9 = 15 counts the same person twice. The row has 14, not 15. From the other end is not total minus the place. A row of 20 does not put the 7th person at 13th from the right. Add 1 back. The two numbers must be the two ends of the same person. Two different people need the between-rule, not this one.

Practice questions

1. A person is 8th from the left and 5th from the right. How many people are in the row?
2. A row has 15 people. What is the place from the left of the person who is 3rd from the right?
3. A person is 11th from the left and 11th from the right. How many people are in the row?

Answers and explanations

1. 12. 8 + 5 - 1 = 12.
2. 13. 15 - 3 + 1 = 13.
3. 21. 11 + 11 - 1 = 21.

Analogy

The person is the shared brick in two walls that meet. Each wall is counted from its own end and includes that brick. Add the two lengths and take the brick back once.

Quick reference

• Same person, both ends: total = left + right - 1.
• Known total: other end = total - this place + 1.
• Do not add the two places and stop. That counts the person twice.
• One straight row only. Not a circle.