Notes for Worded Questions
Introduction
Now that we have seen What's My Age type problems, we will look at some other worded
questions were algebra may be useful as a method of solving.
Other Sum or Total Questions
Any question that tells us the RELATIONSHIP between a bunch of unknowns (like Sally has 5
more than Gina and Lisa has 2 less than Sally etc.) and then also goes on to tell us a SUM, a
TOTAL, can be worked out similarly to our “What's My Age” problems.
Example: I have 3 piles of chocolates, one with Aeros, one with Berty Beatles, and the
third with Crunchies. If I have 2 more Aeors than Berty Beatles, and 5 more Crunchies
than Berty Beatles, and a total of 67 chocolates, how many of each do I have?
Solution:
1. Pick one of the unknowns to be
Number of Berty Beatles = x
2. Represent other unknowns as expressions in terms of x
Number of Aeros = x + 2
Number of Crunchies = x + 5
3. Find an expression for the total
Total = (x) + (x + 2) + (x + 5)
Total = x + x + 2 + x + 5
Total = 3x + 7
4. We know the total is 67, so make an equation with the expression we just found
3x + 7 = 67
5. Solve equation
3x + 7 = 67
3x = 60
x = 20
6. Answer question
Number of Berty Beatles = x = 20.
Number of Aeors = 20 + 2 = 22
Number of Chrunchies = 20 + 5 = 25.
CHECK:
Total = 20 + 22 + 25 = 67. Just what we wanted.
Interestingly, this same problem could have been solved with equal piles.
Equal Piles
If we start with 3 piles, knowing the total is 67.
Take 2 Aeros away so that we have equal amounts of Aero and Berty Beatles. Now the total is
down to 65.
Take 5 Crunchies away so we also have the same amount of Crunchies compared to the others.
Now the total is down to 60.
So now we have 3 equal piles, which add up to 60.
So each pile is 20.
Thus, there are 20 Berty Beatles. Add 2 more and we get 22 Aeros. Add 5 more to 20, and we find
we have 25 Crunchies.
Another Example
There are 3 times as many cows on a farm as sheep. And 5 times as many chickens as sheep. If
there are 99 animals in total, how many of equal animal?
Solution:
1. Define one of the unknowns as 'x'
Number of sheep = x
(I choose sheep for convenience)
2. Come up with expressions for the other unknowns
Number of cows = 3x
Number of chickens = 5x
3. Come up with an expression for the total/sum
Total =(x) + (3x) + (5x)
Total = 9x
4. Come up with an equation using previous expression since total is 99
9x = 99
5. Solve equation
9x = 99
x =11
6. Answer question
Number of sheep = x = 11
Number of cows = 11 * 3 = 33
Number of chickens = 11 * 5 = 55.
Consecutive Numbers
The sum of consecutive numbers problems can be worked out exactly as the ones we have solved
before. The trick is to realise the RELATIONSHIP between consecutive numbers.
For examples:
1,2,3,4 could be written as 1, 1+1, 1+2, 1+3
8,9,10,11 could be written as 8, 8+1, 8+2, 8+3
103, 104, 105, 106 could be written as 103, 103+1, 103+2, 103+3
In general, when you have 4 consecutive numbers, if you call the first number x, then the next
number is x + 1, the one after that x + 2, after that x + 3, etc. etc.
So consecutive numbers are not too bad to deal with in relation to the smallest or first number in
the sequence.
Example:
3 consecutive numbers add to 33.
Solution:
first number = x
next number = x + 1
last number = x + 2
Sum = (x) + (x+1) + (x+2)
Sum = x + x + 1 + x + 2
Sum = 3x + 3
So the equation is 3x +3 = 33.
Solve it:
3x + 3 = 33.
3x = 30.
x = 10.
So numbers are 10, 11, 12.
Consecutive Odd Numbers or Consecutive Even Numbers
Think about consecutive odd numbers.
1, 3, 5, 7 is really just 1, 1 + 2, 1 + 4, 1 + 6
13, 15, 17, 19 is really just 13, 13+2, 13+4, 13+6
Notice you must add 2 more each time to remain odd.
In general, 3 consecutive odd numbers would be x, x + 2, x + 4.
It is the same with even numbers since
2, 4, 6, 8 is just 2, 2+2, 2+4, 2+6
18, 20, 22, 24 is just 18, 18+2, 18+4, 18+6.
In general, 3 consecutive even numbers is also x, x + 2, x + 4.
Word Puzzles Relating to Numbers
Another type of problem that algebra can be useful in solving are word puzzles related to numbers
like “3 times a certain number is 3 more than 15”.
Algebra is useful because the “words” can normally be translated to a simple equation which we
can then solve.
For example:
3 times a certain number is 3 more than 15
Solution:
Well, a certain number can be 'x'.
3 times a certain number would be '3x'.
'is' means =.
3 more than 15 is 18.
Put it all together.
3x = 18
Then solve it.
3x = 18
x = 6.
So the certain number was 6. 3 times 6 is 18. And 18 is 3 more than 15.
A Useful Dictionary
It is sometimes good to keep a list of phrases in your notes book to remind you what some words
actually mean when converted to “maths” symbols. Here are some to get you started.
3 more than a certain number
the sum of a certain number and 6
the sum of 3 consecutive numbers
the difference between a certain number and 5
5 less than a certain number
When 3 is subtracted from a certain number
When a certain number is subtracted from 7
5 times a certain number
3 lots of a certain number
A certain number is multiplied by 6
Twice a certain number
A certain number is divided by 7
Half of a certain number
x+3
x+6
(x) + (x+1) + (x+2)
could be x – 5 or 5 – x
x–5
x–3
7–x
5x
3x
6x
2x
x
7
x
2
A third of a certain number
two thirds of a certain number
2x
3