Astudent solved this problem by working backward.
tom had some marbles. he had three times as many marbles as ralph. ralph had 6 marbles. tom lost 3 marbles and bought 6 new marbles. how many marbles does tom have now?
student's solution - ralph has 6 marbles, so tom has 6 × 3 marbles to begin with. so tom has 18 marbles. he lost 3, so 18 – 3 = 15, and bought 6, so 15 + 6 = 21. so tom now has 21 marbles.
here is a problem that can also be solved by working backward.
alex bought twice as many pears as apples. he bought 4 more pears than oranges. he bought 6 apples.
how many oranges did alex buy?