Forty Problems Forty Problems Forty Problem

Forty Problems for the Classroom

With Hints & Suggestions for further work in Secondary & Tertiary Classrooms Published by and available from

The Association of Teachers of Mathematics

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Forty Harder Problems for the Classroom

Sugar boxes A manufacturer packages his 1 kg bags of sugar in large cardboard boxes. Each box has sides in the ratio 9 : 6 : 4. If the sugar was packed in boxes of the same size but cubical he would save on cardboard. Find the ratio of the areas of cardboard needed for each of the boxes.

Congruent pieces of a triangle Draw a row of equilateral triangles about 4cm high.

Split the first triangle into two congruent triangles and the second into three congruent triangles. Can you split the others into 4, 5, 6 . . . congruent triangles up to 12 or more? Are there some that do not work? Can some of the triangles be split in more than one way? Would any of your methods work starting from one general triangle? Try to find a systematic method to enable you to predict, for example, whether the triangle can be split into 24, or 25, or even 48 or 49 congruent triangles without having to draw the whole figure. Are any of these possible starting from a general triangle?

Divisibility by 9

874539÷9= a, b and c represent the three digits, in that order, of the 3-digit number n. What is the value of n, in terms of a, b, c? Show that if a + b + c is divisible by 9, then n is also divisible by 9. Write down the converse of the statement and find out if it is true. Do you get the same result with a 4-digit number?

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Angles in three squares This rectangle is made up of three squares. Prove that x + y = z.

x

y

z

Note that trigonometry is not necessary.

Leaf area Four semicircles of radius r are drawn inside a square of side 2r so that each semicircle has a side of the square as its diameter. The four-leaf figure formed where the semicircles overlap inside the square is shown shaded. Prove that the area of the shaded figure is 2(π – 2)r².

Moving a digit

? ? ? ? ? 4 A number has 6 digits. The last digit is 4. If the 4 is moved to the front of the number it becomes 4 times greater. What is the number? Another number has six digits and the last digit is 5. If the 5 is moved to the front the number becomes 4 times greater. What is this number? Now try different final digits for the number.

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Billiard table Using squared paper, draw x and y axes meeting at (0,0). A is at (12,0), B is at (12,8) and C is at (0,8). The rectangle 0ABC represents a billiard table without pockets. Treat the ball as a point and assume it bounces off the cushions according to the law of reflection.

a The ball is projected from the point (9, 4) so that it strikes the side 0A first. Find the range within which it must do this so that it strikes the side 0C next. b Suppose it is now projected from the same point to strike 0A at (3, 0). Trace its course as it bounces off successive cushions and find if it passes through its starting point. c Find out what happens if you try another direction of projection from the same point.

Two cars A man has two cars, a Niscom Tiny and a Fita Minute. The ratio of the number of miles each does to the gallon is 5 : 4. The ratio of fuel prices per litre is 3 : 2. The ratio of miles covered per week by each car is 3 : 2. What is the ratio of the total expenses on fuel for the two cars in one week?

Harder bike ride Tom and Jerry are on their bikes again. Tom rides at 10 mph uphill, 15 mph on the level and 20 mph downhill. Jerry rides at 15 mph all the way. If they set off together up into the Pennines, returning by the same route, do they arrive home at the same time?

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numbers in a sequence

…, 1368, 14 4 3, 1520, … In a sequence of numbers the nth number has the value n2 + 2n. Write the first six numbers in the sequence. Explain why the numbers in the sequence alternate between odd and even. How many numbers in the sequence are prime? Explain how you know.

Exam percentages In an examination taken by both boys and girls, the pass rate for boys is 64% and for girls it is 70%. Of the failures 60% are boys. What percentage of the candidates were boys?

Cycling uphill Imagine you cycle up a hill following a regular zig-zag path with a constant gradient and that in this way you can cycle twice as fast as if you go up the line of steepest slope. If you take the same time to cycle to the top by either route, find the angle between each leg of the zig-zag and the line of steepest slope.

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How fast does your hair grow? Estimate, in miles per hour, the speed at which your hair grows.

Squares in circles in squares In a square a circle is inscribed. In the circle another square is drawn and so on for ever. Show that the sum of the areas of all the squares is twice that of the first. Now compare the sum of the areas of all the circles with the area of the first circle.

Fraction of a triangle Find the area of the small triangle as a fraction of the large one.

Given the values of a and b, and provided that b is big enough, there are usually two values of d for which the small triangle is similar to the large triangle. Find the product of these values. How big does b have to be? If b is big enough and there are not two values what is the value of d?

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Sums of odd numbers

1+3+5+…+97+99+101 What is 1 + 3 + 5 + … + 97 + 99 + 101? One way of doing this is to write the sum of the numbers as S. Then S = 1 + 3 + 5 + … + 97 + 99 + 101 and S = 101 + 99 + 97 + … + 5 + 3 + 1 and then think of an efficient way to find the value of 2S. Now find the sum of the first n odd numbers. Now find this sum: 1 + 3 + 7 + 9 + 13 + 15 + … + 393 + 397 + 399.

Shoes and marriage a T here are n people in a crowd. 17% of them are onelegged and half of the remainder are barefoot. Find the number of shoes that are being worn.

b In a village 40% of the males are married to 60% of the females. What percentage of the village population is married?

The length of a drive belt In a model of a piece of machinery two pulleys of radii 2 cm and 7 cm have their centres 10 cm apart. They are connected by a driving belt so that they rotate in the same sense. Assuming that the free parts of the belt are straight, show that its total length is (10√3 + 32π/3) cm.

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Enclosed shapes and their areas You can only use four straight lines 2 cm in length and four that are 1 cm long. Use all of these eight lines to enclose each of the following:

a a capital L as used by learner drivers; b four equal squares; c three equal squares. d the largest possible square; e a regular polygon even larger than the square in (d), and prove that it is larger. f an equiangular polygon even larger than the regular polygon in (e) and prove that this is larger. Try to answer this question without using a calculator.

Divisibility of a formula (1)

( n 5 – 5 n 3 + 4 n ) ÷1 20 = ? If n is any positive integer prove that:

a n(n + 1) is divisible by 2; b n (n + 1) (n + 2) is divisible by 6; c n5 – 5n3 + 4n is divisible by 120. d n5 – 20n3 + 64n is either odd or divisible by 3840.

Width of a border Take a rectangular piece of paper. Along each side cut a border of constant width to leave a smaller rectangle which has an area that is half that of the original rectangle. The problem is to find the width of the border. A method is to add the length and breadth of the rectangle, subtract the diagonal and divide the result by 4. Prove that this method works for all rectangles.

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Two-piece jigsaw Prove by geometry, not by chopping up paper, that two triangles with sides 8, 7 and 5 cm, and 8, 7 and 3 cm, can be fitted together to form either an equilateral triangle or a cyclic quadrilateral. Imagine now that the triangle is divided into a different number of strips of equal height. Under what circumstances would you be able to shade in some of the strips in such a way that you shade exactly half the area? Prove your answer.

Tilting a water trough The cross-section of a v-shaped water trough is an equilateral triangle of side 24 cm. If the trough is tilted about its horizontal axis the water first overflows when one of the sides is vertical. Show that the depth of the water before it was tilted was 6√6 cm.

Height of a bridge The arch of a bridge over a river consists entirely of an arc of a circle of radius r metres. The greatest height of the arch is h metres above the water and the width of the river where it passes under the bridge is 2k metres. Find the relationship between r, h and k. If k = 12 and r = 15 find the value of h. How many solutions are there?

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The sculler A sculler rows upstream for 10 minutes but takes only 5 minutes to return to the starting point. Find the ratio of the sculler’s speed in still water to the speed of the current. Find both speeds if a point one kilometre upstream is reached.

Fruit algebra A bag of 5 apples and 3 bananas costs the same as a bag of 2 apples and 5 bananas. Find the ratio of the price of an apple to the price of a banana. If either bag costs £1.71 find the price of each fruit and check that this agrees with your first result. A bag of 2 lemons, 3 oranges and 4 grapefruit costs the same as a bag of 6 lemons, 2 oranges and 3 grapefruit; and a bag of 8 lemons, 3 oranges and 5 grapefruit costs the same as a bag of 5 lemons, 9 oranges and 2 grapefruit. If a bag containing equal numbers of oranges and lemons costs the same as a bag containing only grapefruit, what is the smallest number of grapefruit there could be in the bag?

Turning circle When a car travels on a curve of fixed radius the whole car rotates about a point on the back axle produced. This is to avoid sideslip of the rear wheels. The steering turns the front wheels through suitable angles about a vertical axis to ensure they also roll without slipping. On a particular car the centres of the wheels form a rectangle 10 feet long (front to rear) and 6 feet wide. The steering is set so that the inner rear wheel describes a circle with a radius of 24 feet. Find: a the radii of the circles described by the other wheels; b t he angle that each of the front wheels makes with its normal position for straight travel; c which wheel goes furthest in a right angle turn of the car.

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