Hands-on math!

How to solve problems by Guess-and-Check

Problems in which you must find the value of 2 or more things can often be solved with the guess-and-check approach.

Here's an example of such a problem:

Jamie went out to her grandfather's farm.
Her grandfather has pigs and chickens on his farm.
She noticed that there were a total of
26 heads and 68 feet among them.
How many chickens and how many pigs
did her grandfather have?

To solve a problem like this, you make a table. The columns (down) of the table are the things you know and the things you want to find out. Each row (across) of your table is a different guess.

Here is the table for the pigs and chickens problem:

Guess
no.		 Number of
pigs		 Number of
chickens		 Total number
of heads		 Total number
of feet
1
2
3
4

You guess the number of pigs and chickens, making their total numbers add up to 26 (the number of heads), then you compute the total number of feet and check!

Let's guess that there were an equal number of pigs and chickens and see what happens:

Guess
no.		 Number of
chickens		 Number of
pigs		 Total number
of heads		 Total number
of feet
1
13
13
13 + 13 = 26
2 x 13 + 4 x 13 = 78 <== too high!

The number of heads is OK, but the number of feet is too high!
(We only needed 68 feet!) Now, here is the key point in solving this problem:
  • If your guess produces a result that is TOO HIGH, then on your next guess you lower the number of pigs (and raise the number of chickens to keep the number of heads the same), because pigs have more feet than chickens.
  • If your guess produces a result that is TOO LOW, then on your next guess you raise the number of pigs (and lower the number of chickens).
So we must LOWER the number of pigs because pigs have more feet than chickens and we have too many feet.

Let's guess 20 chickens and 6 pigs (this keeps the number of heads at 26).
This produces for our second guess:

Guess
no.		 Number of
chickens		 Number of
pigs		 Total number
of heads		 Total number
of feet
1
13
13
26
78
2
20
6
20 + 6 = 26
2 x 20 + 6 x 4 = 64 <== too low!

Well, now the number of feet is too low! So we must RAISE the number of pigs and lower the number of chickens in order to get more feet.

Let's guess 18 chickens and 8 pigs:

Guess
no.		 Number of
chickens		 Number of
pigs		 Total number
of heads		 Total number
of feet
1
13
13
26
78
2
20
6
26
64
3
18
8
18 + 8 = 26
18 x 2 + 8 x 4 = 68 <== the right answer!

Horray! This produces the right number of feet!

So, the key to solving guess-and-check problems is knowing which way to change things when your guess is too high or too low:
  • If your guess is too HIGH, you lower the number of things that have MORE legs or whatever you are computing.
  • If your guess is too LOW, you raise the number of things that have MORE legs or whatever you are computing.
Use as many rows (across) as you need to solve your problem.
For a little practice, here's a slight change in this problem for you to solve:


Jamie went out to her grandfather's farm.
Her grandfather has pigs and chickens on his farm.
She noticed that there were a total of
38 heads and 100 feet among them.
How many chickens and how many pigs did her grandfather have?
Here is a blank table for you to use to solve this problem:
(Use more rows if you need them)

Guess
no.		 Number of
chickens		 Number of
pigs		 Total number
of heads		 Total number
of feet
1
2
3
4