Twice as many is half as less

Sometimes two times as many is half as less.A problem that is a problem over and over goes something like this:

The track team has twice as many members as the cycling team. The cycling team has three times as many people as the water polo team. If a student can only be on one team and there are 100 students on all three teams, how many run track?

Solution:

Twice as many and three times as many both mean multiplication. But students often reserve the order. Let’s take a look at the wrong answer and then a way to remember how to do it correctly.

First let’s set some variables.  Notice that the teams or clubs almost always start with a different letter. I like to use that letter as my variable. Some students work better with always using x & y. That’s fine, just keep them straight.

T = number of students who run track. It is the answer to the question.

C = number of students on the cycling team.

W =  number of students on the water polo team.

Here is the mistake “twice as many on the track team as cycling.” Students often multiply twice the track team.

2T = C

Wrong. I see this algebraic equation written time and again.

Here is a simple memory trick that I teach my students, ask yourself which team has MORE people. 

If there are twice as many on the track team, doesn’t that mean that the track team has MORE people than the cycling team?  If the equation 2T = C is correct then put 10 people on the track team, substituting 10 in the place of the variable T. 

2(10) = C

20 = C, [T]10 is less than [C]20 meaning that the track has LESS people.

Twice as many on track means that there is 2C for every one T.

T= 2C; Track (T) Has (=) Twice (times 2) Cycling (C) .

The way to keep your translations straight is to do as I did above and put in a basic number like 10 and see which one has more, the one with twice is always more. 12 often works since it is the first number that has more than four factors, hence the reason that a dozen is still a highly popular base unit in baking and in inches.

If track has twice as many than cycling, than cycling has HALF as many as track.

T = 2C or T/2 = C

Cycling has three times as many as water polo. Cycling has MORE people.

C = 3W

Put 12 people on the cycling (three times as many, use a multiple of three).

12 = 3W

4 = W, we’re ok because 4 is LESS than 12. 12 is three times as many as 4.

If cycling has three times as many as water polo then water polo has one-third as many as cycling

C = 3W and C/3 = W

The total number of students is 100.

T + C + W = 100.

If we want to know how many run track, then we have to put the other variables “in terms of”  T.

C = T/2, one done.

W = C/3 W is in terms of C now substitute C in terms of T.

W = [T/2]/3

Simplify

W = T/6

Substitute for C and W

T + T/2 + T/6 = 100.

Find the common denominator, which is 6.

6T/6 + 3T/6 + 1/6 = 100

Add up the fractions:

10T/6 = 100

Multiply both sides by 6/10 to find the value of one T

(6/10)(10T/6) = (100)(6/10)

T = 60

Let’s check our work

If twice as many people run track than cycle, then there are 30 cyclers. If there are three times as many cyclers as water polo players then there are 10 polo players. Added up 60 + 30 + 10 = 100 total. Check.