Unit interval
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
Age
14 to 18
Challenge level
Problem
Take any two numbers between $0$ and $1$. Prove that the sum of the numbers is always less than one plus their product.
That is, if $0< x< 1$ and $0< y< 1$ then prove
$$x+y< 1+xy$$
Getting Started
If you know $0< x< 1$ what can you say about $1-x$?
What can you say about the product of two positive numbers?
Student Solutions
Given any two numbers between $0$ and $1$ you have to prove that their sum is less than 1 plus their product; that is, given $0 < x < 1$ and $0 < y < 1$, prove that $x + y < 1 + xy$.
Hyeyoun Chung, St Paul's Girls' School, and Andaleeb Ahmed, Woodhouse Sixth Form College, London both produced nice solutions.
Consider $1-x$ and $1-y$. Since $0 < x < 1$ and $0 < y < 1$ it follows that
$\begin{eqnarray} \\ (1 - x)(1 - y) & > & 0 \\ 1 - x - y + xy & > & 0 \\ 1 + xy & > & x + y. \end{eqnarray}$
This is equivalent to $x + y < 1 + xy$.
Teachers' Resources
Why do this problem?
The problem gives practice in solving linear and quadratic inequalities.
Possible approach
Use this as a lesson starter. If learners do not know how to start let them use the Hint.
Then discuss the problem as a class but try to elicit ideas from the learners themselves.
Key questions
How do we make use of the information given?
If we are not making progress, then have we used all the information given?