Astudent solved this problem by working backwards.

joani has some stuffed animals. she has 2 times as many as lily. lily has 6 stuffed animals. joanie gave 2 away and bought 4 more. how many stuffed animals does joani have now?

students's solution: lily has 6 stuffed animals, so joani had 6 × 2 to begin with. so joani had 12 stuffed animals. she gave away 2, so 12 – 2 = 10. she bought 4, so 10 + 4 = 14. so joani has 14 stuffed animals.

here is another problem that can also be solved by working backwards.

wyatt has 3 times as many baseballs as golf balls. he has 5 fewer tennis balls than baseballs. wyatt has 4 golf balls.

how many tennis balls does wyatt have?

Mr. brettler is hanging up a 5-foot tall rectangular welcome sign in the gym. he has drawn a sketch on a coordinate grid. he has the lower vertices of the sign at (9, –3) and (–6, –3). where would the upper vertices be on the grid?

Martha has a deck of cards. she has lost some of the cards, and now the deck only contains nine spades, eleven diamonds, eight clubs, and twelve hearts. martha predicts that whenever she draws a card from the deck without looking, she will draw a club one-fifth of the time. which activity would best allow martha to test her prediction? a. randomly draw a card from the box and see if it is a club. b. randomly draw a card. then, continue to draw another card until all eight clubs are drawn. c. randomly draw and replace a card 120 times. then, observe how close to 30 times a club is drawn. d. randomly draw and replace a card 100 times. then, observe how close to 20 times a club is drawn.

Two years ago a man was six times as old as his daughter. in 18 years he will be twice as old as his daughter. determine their present ages.