$6.23 \times 5.4 = 33.642$ write down two more multiplications with the answer of $33.642$

Question

For some reason I’m really stuck on how to work this out. I don’t know whether I’m just over complicating it for myself but I genuinely cannot for the life of me think of how to get the answers

Answer 1

Just factor $623\times54=7\times89\times2\times3^3$. Now you can divide this factorization into subsets, for example $(2\times89)(3^3\times7)=178\times189$ for a result of $17.8\times1.89=33.642$. Or $(3\times89)(2\times3^2\times7)=267\times126$ so we have $2.67\times12.6=33.642$.

Answer 2

If you have $xy = z$ then in general $(x-a)(y+b) = z -ay+bx -ab $

which equals $z$ whenever $bx = ay+ab$ i.e whenever $a = \frac{bx}{y+b}$.

E.g if $6.23 \times 5.4 = 33.642$ taking $b = 0.6$ gives $a = \frac{0.6 \times 6.23}{5.4 + 0.6}= \frac{3.738}{6.0}=0.623$

and $(6.23-0.623)\times(5.4+0.6)=33.642$