Word Problems: Example #3

Word problems can be easy to solve. However, they are considered the most difficult problems. Below is an example of how to easily translate and solve a word problem. It was asked by one of the members and seemed to be a very difficult one.

Note: I do not use "variables" on purpose. It is much more intuitive to use the actual names of quantities, and it is easier to manipulate and understand the solution since you don't have to re-translate in your mind every time you have new information or you find a solution to one of the variables.

Sentence 1: At a concert there are twice as many Adults as Children.
Sentence 1 changed: Adults are twice as many as Children.

Translation 1: Adults = 2xChildren. [or 2Children]

Sentence 2: At intermission 530 Adults go out;

This require a new name: AdultsLeft:= the adults who are left after the 530 go out.

Translation 2:Adults-530 = AdultsLeft

Note: I wrote the translation in the way we think, not in the way we tend to learn to write math, which is not necessary. We DON'T have to write the AdultsLeft name (variable) on the left side of the equation. The way we think is: We had "Adults" and then 530 left, so we had some "new amount of adults" and I translate those concept SIMPLY AND DIRECTLY to paper. I actually don't do any thinking! The only thing I did was to use the name "AdultsLeft" instead of the "new amount of adults" idea but I could even use that and it would work well :-)

Translation 2b: we can rewrite the above as -
Adults = AdultsLeft+530

We don't know if it helps us but this way we can see it in two ways. Sometimes one way happens to fit the question better in terms of relating it to other pieces of information. So don't say, "I'll see which one later" but do it when you can, even you don't see the relevance.

Do what you can: we can exchange Adults from translation 2b into translation 1. We could also exchange Adults from translation 1 into Adults in translation 2. It doesn't matter which one you use since both are using sentence 1 and sentence 2.

Exchange 1: 2Children= AdultsLeft+530

Sentence 2: 25 Children leave

This require a new name: ChildrenLeft:= the children who are left after the 25 go out.

Translation 3: Children-25 = ChildrenLeft
Translation 3b:Children =ChildrenLeft+25

Again we can use it with Translation 1, but we are better off using it with Exchange 1 because Exchange 1 was a later step that used Translation 1 & Translation 2.

Exchange 2:
2x(ChildrenLeft+25) = AdultsLeft+530

Simplify:
2xChildrenLeft+50 = AdultsLeft+530

Simplify:
2xChildrenLeft= AdultsLeft+480 [subtract 50 from each side]

Sentence 4: This leaves twice as many children as adults in the concert hall.

Sentence 4 rewritten : Children left are twice as many as adults left .

Translation 2: ChildrenLeft = 2AdultsLeft

We can use it with Exchange 2 by exchanging the ChildrenLeft quantities.
I highlighted in red the exchange.

Exchange 3:
2x2AdultsLeft= AdultsLeft+480

Simplify:
4AdultsLeft= AdultsLeft+480

Simplify:
3AdultsLeft=480
Simplify:
AdultsLeft=160

Use back by exchanging where possible:
Into translation 2: ChildrenLeft = 2x160 = 320
Into translation 3b :Children = 320+25=34
Into translation 2b: Adults = 160+530 = 690

We found out all the possible Names (variables) and therefore we don't need to do more simplifications and exchanges. But, we want to check whether our numbers work out. Always do this step!

Before intermission:
Children = 345
Adults =690

Indeed, the number of adults is twice the number of children, as in sentence 1.

After intermission:
ChildrenLeft = 320
AdultsLeft = 160

Indeed, the number of children is twice the number of adults, as in sentence 4.

Given that it is all working out, we can move to read the next sentence.

Sentence 5: How many people are at the concert?

This is a bit unclear as to whether they mean "are" as in "before intermission" or as in "after intermission" since the "are" is in the present tense and the whole story is in the present/past tense mixed [ah..writing questions well and clearly is critical as well]. I provide both interpretations and as a teacher would accept either. In fact, regardless of what the question asks we want to check both things.

People = Adults+Children

Before Intermission: Adults + Children = 690+345 =1035
After Intermissiion: AdultsLeft+ChildrenLeft = 160+320 = 480

Of course, all that would take a few minutes without all the complex writing I've done to explain how to solve it in general. Once you get used to this kind of approach, it becomes VERY easy to do solve any type of problem at what ever level because they all become a series of one step problems [translate, simplify, exchange for each new piece of information].