All the Excellence Questions (and more) 2002-2010
Some of the following are made up from Merit questions where the equation was given, and then the question
was to solve it (Excellence would require forming and solving). Sometimes these lead to equations which are
slightly easier than would be expected at Excellence
1.
Mathsville School has two square playing fields.
One playing field is 12 metres wider than the other.
The total area of the two playing fields is 584 square metres.
Form and solve at least one equation to find the width of both the playing fields.
2.
x2 + ax – 8 = (x + b)(x + c)
where a is a whole number greater than 0, and b and c are both integers.
Give all possible values for a.
3.
Emma has an 8 m long piece of rope that she uses to make the circumference of a circle.
C = 2π r, A = π r 2
(i) Calculate the area of the circle.
(ii) George cuts x metres off the 8 m rope and then makes the circumferences of TWO circles, one from
each piece of rope.
Write an expression for the sum of the areas of the two circles in its simplest form.
4.
The diagram shows a square of side length c inside another square of side length a + b.
The area of the large square can be written as (a + b)2 or c2 + 2ab (the area of the small square plus the four
triangles).
Use this information to prove Pythagoras’ theorem for a right-angled triangle.
5.
Peg makes a patchwork rug by sewing small equilateral triangles together to form a pattern. Her
pattern uses rows of equilateral triangles, as shown in the diagram.
There are n rows of equilateral triangle patches to make up the pattern.
(i) Give the formula to find the number of triangles, T, in row n of the pattern.
(ii) Write an expression to find P, the total number of equilateral triangles used to make the pattern in terms of
n, the number of rows.
(iii) Use the expression from part (ii) to calculate the number of rows in Peg’s pattern when she has used a
total of 323 equilateral triangles.
6.
Jim needs to make a path from the front to the back of his house, as shown in the diagram.
x is the width of his path, in metres.
Jim has sufficient concrete to make a path with a total area of 9 m2.
Find the width of the path around his house.
7.
Sheffield school uses two vans to take a group of students on a field trip.
• If two students moved from van A to van B, then the two vans would have the same number of students in
each.
• If, instead, two students moved from van B to van A, then van B would have half the number of students that
were then in van A.
Use this information to find the total number of students on the field trip.
You must give at least one equation that you use in solving the problem.
8.
The sides of a square warehouse are extended by 2 m and 3 m as shown in the diagram.
The area of the extended warehouse is 156 m2.
Find the length of one side of the original warehouse.
9.
Alison is using hexagonal tiles to make patterns.
Alison has 461 hexagonal tiles.
She wants to use all of the tiles to make a pattern like those in the diagrams above.
Write an equation to show the relationship between the pattern number, n, and the number of tiles used, T.
Solve this equation to find the pattern number that would have 461 tiles.
You must:
write an equation,
solve the equation,
write down the pattern number with 461 tiles.
10. Paul bought some CDs in a sale.
He bought four times as many popular CDs as classical CDs.
The popular CDs, P, were $1.50 each.
The classical CDs, C, were 50 cents each.
He spent $52 altogether.
Form and solve a pair of equations to find out how many classical CDs Paul bought.
11. James is five years old now and Emma is four years older.
Form a relevant equation and use it to find out how many years it will take until James’s and
Emma’s ages in years, multiplied together, make 725 years.
12. The diagram shows a design made with 5 rows of small squares
.
The number of small squares, S, in the design is given by a formula, in terms of R, where R is the number of
rows of small squares in the design.
Calculate the number of small squares, S, for a similar design that has 11 rows of small squares, by first finding
the formula for R.
13. The diagram shows a square courtyard with a square pool in one corner.
The area of the courtyard is 225 m2, and the courtyard extends 8 m beyond the pool.
Form and solve an equation to find x, the length of the side of the pool.
14. Josh’s mum gives him $76 to use when he phones her during the year.
Josh wants to have at least $30 left at the end of July.
Each phone call costs him $3.75.
Form and solve an inequality to find out the maximum number of phone calls Josh can make before the end of
July, where x is the number of phone calls.
15. Josh bought mini pizzas for his flat one night.
The Supreme pizza was 50 cents more than the Hawaiian pizza.
The total cost for two Hawaiian pizzas and three Supreme pizzas was $20.75.
Form and solve a pair of simultaneous equations to find the price of one Supreme pizza.
16. At his flat, Josh makes two rectangular gardens.
The herb garden is 2 metres longer than it is wide and has an area of 11.25 m 2.The vegetable garden is 3
metres longer than it is wide and has an area of 13.75 m2.
The combined width of both gardens is 5 metres.
Find the length and width of each garden. State any equations you need to use.
17. A shop sells packs of gift cards in boxes that have clear plastic lids.
Each box is 4 cm high and has a square base.
The outside of each box is painted.
The area painted on each box is 260 cm2.
Form and solve an equation to calculate the length of the base of each box.
18.
In a sale at the bookshop, all the paperback books were one price and all the magazines were another price.
Robyn bought 2 paperback books and 4 magazines and paid $39.70.
Kirsty bought 3 paperback books and 1 magazine and paid $39.80.
Form and solve a pair of simultaneous equations to calculate the sale price of one magazine.
19.
The Eagle Courier Company has a limit on the size of parcels it will deliver.
The size of the parcel is calculated by finding the sum of its length and the distance around the parcel, as
shown by the dotted line in the diagram.
The maximum size of parcel that Eagle Courier Company will deliver has a sum of 210 cm.
A particular parcel is twice as long as it is wide and three times as wide as it is thick.
By forming an equation or inequation, calculate the largest possible dimensions in centimetres of this parcel, if
it is to meet the 210 cm size restriction described above.
20. Mr Smith spends $1210 on 1390 square tiles to use on the bathroom floor.
He buys S small tiles for 80 cents each and B big tiles for $1.50 each.
Form and solve a pair of simultaneous equations to find the number of tiles of each size that Mr Smith
bought.
21. One integer is 5 more than twice another integer.
The squares of these two integers have a difference of 312.
Write at least ONE equation to describe this situation, and use it to find the TWO integers.
22. Janet bought tickets to the diving and the swimming at the Olympic Games.
She paid $1095 for 15 tickets.
The tickets for the diving cost $65 and the tickets for the swimming cost $85.
Form and solve a pair of simultaneous equations to find the number of tickets Janet bought for the
swimming.
23. At the Olympic Games 40 years ago, the average number of competitors per sport was 5 times the number
of sports played.
In 2004 there were 10 more sports than there were 40 years ago.
In 2004 the average number of competitors per sport was 3.5 times greater than 40 years ago.
At the 2004 Olympic Games there were 10 500 competitors.
Write at least ONE equation to model this situation.
Use the model to find the number of sports played at the Olympic Games 40 years ago.