Puzzles
Sum equals product
\(3\) and \(1.5\) are a special pair of numbers, as \(3+1.5=4.5\)
and
\(3\times 1.5=4.5\) so \(3+1.5=3\times 1.5\).
Given a number \(a\), can you find a number \(b\) such that
\(a+b=a\times b\)?
Show answer & extension
Hide answer & extension
If \(a+b=a\times b\), then:
$$ab-b=a$$
$$b(a-1)=a$$
$$b=\frac{a}{a-1}$$
This will work for any \(a\not=1\) (\(a=1\) will not work as this
will
mean
division by zero).
Extension
(i) Given a number \(a\), can you find a number \(b\) such that
\(b-a=\frac{b}{a}\)?
(ii) Given a number \(a\), can you find a number \(b\) such that
\(b-a=\frac{a}{b}\)?
(iii) Given a number \(a\), can you find a number \(b\) such that
\(a-b=\frac{b}{a}\)?
(iv) Given a number \(a\), can you find a number \(b\) such that
\(a-b=\frac{a}{b}\)?
