Problem Solving. Consider Special Cases

When we are dealing with a large number range (e.g. from 1 to 1000), it is often useful to consider what happens with simple cases (e.g. small numbers like 1 and 2) and extreme cases (e.g. 1000) first. This can help us understand the rest of the numbers.

If we have a question related to a specific large value (e.g. 625), we may decide to tackle the question for a smaller value first to see what happens. It will often be helpful if that smaller values shares properties with the larger value (e.g. number 5 because it is also divisible by 5).

Thinking geometrically, we could reduced a problem that is about triangles in general to if it was about only equilateral triangles or only right-angled triangles and then try to generalise our results.

We can also consider the extremes of the situation a problem refers to, such as considering the largest possible number or angle for which the problem applies.

Worked Examples

First we will read both examples and have a quick think about them and then we will look at how considering special cases can help us with each one:

Exams Example

Roger has six exams to complete, each of which will be marked out of 100. He wants to get a mean result of at least 85 for the year. What is the lowest mark out of 100 that he can score in a single test.

Triangles in Circle Example

In the below diagram, O is the centre of the circle, C is on the circumference and AB is a diameter. What is the relationship between the areas of triangles OAC and OBC?

Worked Solutions to Examples

Exams Example

Solution

We start by considering the lowest acceptable mean result, i.e. 85.

To get a mean result of 85 in the six tests would mean a total score of 6 x 85 = 510.

We then consider the maximum she could get on five of the tests, i.e. 100 on each, so 5 x 100 = 500.

Then we can see by comparing these that the lowest possible score she could get on a single test is 510 – 500 = 10.

Triangles in Circle Example

In the below diagram, O is the centre of the circle, C is on the circumference and AB is a diameter. What is the relationship between the areas of triangles OAC and OBC?

Solution

At first glance, it looks like the areas are very similar or possibly the same.

Without changing the conditions set by the question, we could consider the special case where OC is perpendicular to AB, as shown below:

We can see by symmetry that in this case the two triangles have an equal area.

Now let’s consider the opposite (almost silly seeming) extreme, where C is exactly at point B, at one extreme of the diameter. In this case (shown below) both of the triangles have a zero area, so they again have an equal area:

So it starts to seem more convincing that the two triangles have the same area. Now, by considering the formula for the area of a triangle: 1/2 x base x height, we can use the below diagram to convince ourselves that is true for all cases:

So, thinking about the formula, first we notice that the base lengths of the two triangles are both the same (i.e. AO and OB are both radii of the circle). We then notice that the triangles both have the same height (this is marked on the diagram above as a dotted line). So as all of the variables in the formula are the same for each triangle, the triangles must have the same area.

19 Questions of Increasing Difficulty

(1.*) In a restaurant, chairs can be arranged around separate tables, as shown in the diagram:

The restaurant owner has a total of 55 chairs.

(a.) How many chairs are needed if 5 tables are arranged as demonstrated in the last diagram above?

(b.) Find the rule for the number of chairs needed for n tables arranged as demonstrated in the last diagram above?

(c.) Is it possible to use all 55 chairs with tables arranged like in the last diagram? Explain your answer?

The restaurant needs to be prepared for 32 guests at a wedding reception. The bride wants the guests to be split into two equal groups, so they will sit at two large tables.

(d.) How many small square tables will be used to make each of the large tables at the reception?

(2.* – No calculator) Dasha is asked to choose five different simplified fractions and to write them down in ascending order. She thinks carefully and then writes:

$\frac{1}{6} , \frac{1}{5} , \frac{1}{4} , \frac{1}{3} , \frac{1}{2}$

(a.) Suggest another set of fractions that she could have written;

(b.) Suggest another set of fractions where none has a numerator of 1;

(c.) Suggest another set of fractions, all of which have their numerator greater than the denominaotr.

(3.*) Which is greater in each of the following pairs?

(a.) 400g + 400mg or 0.5kg + 90g;

(b.) 0.1km + 150cm or 110m + 900cm;

(c.) 0.75 hours + 600 seconds or 50 minutes + 0.1 hours;

(4.*) Masha has a bag of coloured balls. Each ball is either red, green or yellow. The probability of picking a red ball at random is 3/7. The probability of picking a yellow ball at random is the same as the probability of picking a green ball.

(a.) There are nine red balls. How many balls are there in total?

Masha adds more red and yellow balls to the bag. The probability of picking a green ball at random is now 3/25.

(b.) What is the total number of balls in Masha’s bag now?

(5.*) On average there are over 100,000 strands of hair on a child’s head. Blondes average about 140,000 strands, brunettes average 108,000 strands, and and redheads average 90,000 strands. Hair grows at a rate of about 150mm per year. The average person loses about 0.25 per cent of their hair strands each year.

If hair does not grow back,

(a.) Approximately what percentage of their hair strands would redheads lose after 82 years?

(b.) How many years would it take for an average blonde to lose one tenth of their hair strands?

(6.* No calculator) How can you tell that 23.64 x 805 = 1903.02 is incorrect?

(7.**) I choose three consecutive integers and add them together: 5 + 6 + 7 = 18. I notice that 18 is also the result of 6 x 3.

I choose another set of three consecutive integers and add them together: 2 + 3 + 4 = 9. I notice that 9 is also the result of 3 x 3.

It looks as if the sum of three consecutive numbers is the same as the middle number multiplied by three.

(a.) Is this always true?

(b.) Can you find a rule for adding four consecutive integers?

(c.) How might you extend your rule to add five or more consecutive integers?

(8.**) Four snails have a race. Daddy snail travels at 36,000 mm/h. Mummy snail travels at 0.01 m/s. Baby snail travels at 5km/day. Auntie snail travels at 700cm/h. Assuming that the snails all remain at a constant speed, which will arrive at the finish line first?

(9. **) The rabbits in a zoo are kept in a circular enclosure with diameter 20m. It has a fence around it. It then has a second circular fence around that one, leaving a gap of 2m between the fences.

(a.) How much longer is the outside fence than the inside fence?

(b.) Suppose that the gap was changed to 3m. What then would be the difference between the lengths of the fences?

(c.) What is the general rule?

(d.) What happens if the enclosures are square or rectangular in shape?

(10.**) Sasha draws a shape on a coordinate grid as shown below:

She notices that the centre of her shape, marked with a black dot on the diagram, lies on the line x = -5.

(a.) How could Sasha translate her shape so that is centre remains on the line x=-5? (we should use vector notation to describe translations).

Sasha’s shape also lies on the line y = x.

(b.) How could you translate the shape so that its centre remains on the line y=x?

(c.) How could you translate the shape so that its centre lies on the line y=x+3?

(11.**) The diagram below shows two cereal boxes:

Kiefer says: “I think that the Choc Flakes box has the greater volume”.

Arpad says: “I think that the BioWheat box has the greater surface area”.

Raymon says: “Only one of you can be correct. Whichever box has the greater volume must also have the greater surface area”.

Who is correct? Justify your answer.

(12.**) Micah is asked to investigate what happens when a shape is reflected in two perpendicular mirror lines, one after the other.

He draws the following picture and notices that the result is the same as if he had translated the original shape.

Will this always happen? What can Micah do to check if it will?

(13.**) Melissa has designed the following logo for her business:

It is based on an isosceles triangle with a semicircle on each of its sides. Calculate the perimeter of Melissa’s logo, giving your answer to 2 decimal places.

(14.**) Calculate the mean average of the following three numbers: $\sqrt{24} , \sqrt{54} , \sqrt{96}$

(15.***) Each of the seven small triangles in the diagram below is an isosceles triangle. The large outer triangle is also isosceles:

Given one angle, can you work out all of the others?

Ingrid says: “No, you don’t have enough information”;

Helen says: “Yes, you can work out all of the other angles”;

Pete says: “I wonder what would happen to the other angles if the first one was 10° instead of 12°

(a) Decide whether Ingrid or Helen is correct, being sure to justify your answer.

(b.) Try Pete’s idea. What would happen if we started with an angle of 10°?

(c.) Is there another startingangle for which this will work?

(16.***)

(a) What’s the same and what’s different about the following fractions and their decimal equivalents:

$\frac{1}{9} , \frac{1}{99} , \frac{1}{999} , \frac{1}{9999} , \frac{1}{99999} , \frac{1}{999999} ,$

(b – No calculator) Explain how you can use your findings to work out the equivalent fraction to each of these recurring decimals:

(i.) $0. \dot{7}$

(ii.) $5. \dot{3}$

(iii.) $0.1 \dot{4}$

(iv) (i.) $2.5 \dot{0} 00 \dot{9}$

(17.***)

Jack holds out a coin and tries to cover the Moon. He can’t make it fit exactly – the coin is too big. He gets a friend to hold the coin steady and moves backwards until the coin is a good fit over the moon. He finds that he has to walk about 2.5m away from the coin.

Given that the Moon has a diameter of 3474.8km and that the coin has a diameter of 22.5mm, use Jack’s findings to estimate the distance of the moon from the earth.

(18.***) Solve the equation $(2^x)^2 - 20(2^x) + 64 = 0$