The Lady's Age

The Problem

Mathematical curiosities and puzzles have fascinated people throughout the ages. These were often expressed in verse or as riddles. Here is one of these.

A lady being asked her age by an impertinent young spark, gave the following reply:

If to my age there added be;
One half, one third and three times three
Six score and ten the sum you’d see
Now pray tell what my age may be?

[A Lady’s Diary, 1788]

Teaching sequence

1. Talk about historical things. (If you can find some relevant pictures to show the class, so much the better.)
   Who is the most famous person you know who was born over 50 years ago?
   When was the Treaty Of Waitangi signed?
   When was the Eighteenth Century?
   Can you tell us of something that happened in the Eighteenth Century?
2. Tell the class the riddle from the Lady’s Diary. Make sure they have some idea of what it is about.
   Why do you think the problem was invented?
   What do you think the lady who wrote it looked like?
   What did she wear?
   What sort of house did she live in?
   What is a ‘score’? What is ‘six score and ten’?
   What does ‘one half’ and ‘one third’ refer to? (The lady’s age.)
3. Get the class to work on the problem in groups of two or four.
4. Circulate around the class and check on progress. If a group has finished using an algebraic approach, then let them try the Extension problem.
5. Allow time for several groups to report on their answer and the ways that they solved the problem. If there is not time to look at the Extension problem ask them to take it home and get their parents’ help.
6. At some stage let the class write up two ways of solving the problem in their mathematics book.

Extension to the problem

Can you make up a problem about your own age or about someone else’s? Give it to another member of the class to solve.

Use the Internet to find books that contain mathematical problems, puzzles or recreations. Don’t be restricted to Western literature, you might find some references to Chinese and Japanese writing. Let us know of anything interesting that you find and we’ll add it to this problem on this web site.

Solution

As in a number of other problems, we give three possible approaches to this problem. All of you students should be able to use Method 1. The more sophisticated method is Method 3 where we use algebra. Hopefully most of your class can at least get started on the algebraic approach.

Method 1. Guess and Improve. We could try this problem by Guess and Check but that is very inefficient. So instead we’ll try Guess and Improve. We want to make an initial guess and then use that guess to get a better one and gradually hone in on the answer.

Now three score and ten is 130, so we have to guess an age for the lady so that it plus one half of it plus one third of it plus three times three equals 130.

Suppose the lady was 48. Then one half of 48 is 24 and one third of 48 is 16. Clearly three times three is 9. So we have 48 + 24 + 16 + 9 = 97. This is too small. So try a bigger number.

Suppose she was 90. Then 90 + 45 + 30 + 9 = 174 - too big. So we have to try something bigger than 48 and smaller than 90.

Suppose she was 60. Then 60 + 30 + 20 + 9 = 119 - too small. So we want a number between 60 and 90.

Suppose she was 66. Then 66 + 33 + 22 + 9 = 130: AHA!!

Method 2. Test every possible combination. This is a rather laborious way of doing the problem. However, if you have someone who is good at writing computer programs, then you will get the calculations done very efficiently.

The program might run through all numbers from 1 upwards checking the criteria of the problem.

Method 3. Use algebra. Let the ladies age be a. (We think that it is a good thing to use a letter that has some relevance to the problem. Of course there is nothing wrong with x but what is the link between x and age?)

Then we have to solve So Hence This gives or Tidying up we get

a + a/2 + a/3 + 9 =  a + a/2 + a/3 =  (6a + 3a + 2a)/6 =  11a/6 = a = a =

130. 121. 121. 121 (121 x 6)/11. 66

This method is a lot more efficient than the other two. This is often the case with algebra. It’s really quite a powerful technique.