We don’t always need to draw a picture. Often, a table can be just as help- ful. In the bookshelf example, we drew six shelves and placed 16 books on each shelf. We could have drawn a table and placed the numbers of books in the table:
Shelf Number Number of Books 1 16 2 16 3 16 4 16 5 16 6 16
The table, just like the picture, helps us to see that we have six sets of 16. The best way to find the total number of books is to multiply 6 by 16: 6 ≈ 16 = 96 books.
The type of table we build depends on the information in the word prob- lem. Tables are especially good for organizing multiplication problems, as we just saw. But they are also good for solving other kinds of problems. Example
A carnival game has red pins and blue pins. For every red pin that is knocked down, a player scores three points, and for every blue pin that is knocked down, a player scores five points. Fred knocked down eight pins and scored 34 points. How many of each pin did Fred knock down?
We can tell right away that this isn’t a typical addition, subtraction, mul- tiplication, or division problem. We know that Fred knocked down eight pins, so the number of red pins plus the number of blue pins must equal eight. We also know that Fred scored 34 points, so the number of points scored from red pins plus the number of points scored from blue pins must equal 34.
Let’s say Fred knocked down only red pins. Since each red pin is worth three points, Fred would have scored 8 ≈ 3 = 24 points. But Fred scored 34 points, so he must have knocked down some blue pins. By putting the infor- mation in a table, we can make different combinations of red and blue pins until we find the right total. The number of red pins times three plus the number of blue pins times five must equal 34. The following table shows all the possible combinations of eight pins that Fred could have knocked down:
Number of Points Number of Points from Red Pins from Blue Pins Number of Number of (number of red (number of blue
Red Pins Blue Pins pins times 3) pins times 5) Total
8 0 24 0 24 7 1 21 5 26 6 2 18 10 28 5 3 15 15 30 4 4 12 20 32 3 5 9 25 34 2 6 6 30 36 1 7 3 35 28 0 8 0 40 40
The sixth row of the table shows Fred scoring a total of 34 points. To score 34 points, he had to knock down three red pins and five blue pins. Solving this problem without a table would have been tough!
INSIDE TRACK
WHEN BUILDING Atable, you can stop filling in numbers once you have found the answer. In the last example, we could have stopped after completing row 6. Remember, the table is a tool for helping you find the answer; completing the table is not the goal.
Example
Ice cream cones cost $2, and ice cream cups cost $3. Dom sells a total of ten cones and cups and collects $23. How many cones and cups did Dom sell?
We can build a table similar to the one we built in the last example. The amount of money Dom collected is equal to the number of cones sold mul- tiplied by $2 plus the number of cups sold multiplied by $3. If Dom sold only cones, he would have collected 10 ≈ $2 = $20, and if he had sold only cups,
he would have collected 10 ≈ $3 = $30. Since Dom collected $23, he must have sold some of each:
Money Money Number Number Collected from Collected from
of Cones of Cups Selling Cones Selling Cups Total
10 0 $20 0 $20
9 1 $18 $3 $21
8 2 $16 $6 $22
7 3 $14 $9 $23
We can stop building our table—we’ve found our answer. In order to col- lect $23 from ten cones and cups, Dom must have sold seven ice cream cones and three ice cream cups.
Some word problems don’t involve numbers at all. Building a table can help us solve these types of problems, too.
Example
Nicholas is older than Anthony but younger than Louis. Marie is older than Louis but younger than Jill. Who is the oldest, and who is the youngest?
Build a table ordered from oldest to youngest from top to bottom. Start by placing Nicholas in the middle of the table with Anthony beneath him and Louis above him:
Louis Nicholas Anthony
Next, add Marie to the table above Louis, and place Jill above Marie, since Jill is older than Marie:
Jill Marie Louis Nicholas Anthony
It’s easy to see now that Jill is the oldest and Anthony is the youngest. The table we used to solve this problem was simple, but it helped us organize the information given in the word problem.
PRACTICE L AP
11.Rico has seven coins in his pocket, all of which are nickels or quarters. If Rico has $0.95 in his pocket, how many nickels and how many quarters are in his pocket?
12.Tickets to a movie cost $4 for children and $7 for adults. If 15 people see the movie and $87 is paid, how many children and how many adults saw the movie?
13.Five people are standing in line. Katie is ahead of Danny, who is ahead of Kenny but behind Casey. Coral is behind Kenny. If Casey is not first in line, who is?
14.There are six teams in the Eastern Volleyball League. The Saints have 16 wins, the Cardinals have 20 wins, the Comets have eight wins, the Stars have 12 wins, the Bees have 11 wins, and the Tigers have 18 wins. How many games must the Bees win to tie for third place?
15.Harold is taller than Mickey but shorter than Peter. Peter is taller than Joanna but shorter than Dennis. If Joanna is taller than Mickey, which two people could be the same height?
PACE YOURSELF
MAKE A TABLEwith the heights and ages of all your friends. Which one of your friends is the oldest? The tallest? What is the difference in age between your oldest friend and your youngest friend? Your tallest friend and your shortest friend? Use your table to answer these questions.