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Problem-solving Pack
MathematicsYear 8 · 8.8 Equations
Problem-solving Pack
Name: _________________________________
Date: _________________ Class: ___________
These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.
1Problem 1 of 12
Linear-equation grand tour. Solve each, showing your steps:
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
Working space
2Problem 2 of 12
Kite perimeter algebra. A kite has two long sides of length cm and two short sides of length cm. The perimeter is 38 cm.
(a) Set up an equation in and solve.
(b) Find the lengths.
(c) Verify the perimeter.
(a) Set up an equation in and solve.
(b) Find the lengths.
(c) Verify the perimeter.
Working space
3Problem 3 of 12
Songs equation. Pete has 3 times as many songs as Sabrina. Sabrina has 120 fewer than Elena. Pete has 300 songs.
Find Elena's count.
Find Elena's count.
Working space
4Problem 4 of 12
Rectangle algebra. In a rectangle, the long sides are and cm; the short sides are and cm.
(a) Solve for .
(b) Solve for .
(c) Find the perimeter.
(a) Solve for .
(b) Solve for .
(c) Find the perimeter.
Working space
5Problem 5 of 12
Reverse-percentage equation. A bike on sale for 590 chf is at a 26% discount.
(a) Set up an equation and find the original price.
(b) The shop raises the sale price by 26%. Show the new price is not the original, and find the difference.
(a) Set up an equation and find the original price.
(b) The shop raises the sale price by 26%. Show the new price is not the original, and find the difference.
Working space
6Problem 6 of 12
Two-step word problem. I think of a number. I treble it, subtract 7, then halve the result. I get 4. What is my number?
Working space
7Problem 7 of 12
Pricing strategy. Tickets cost chf each. A booking fee of 5 chf is added per order. A school orders 25 tickets and pays a total of 380 chf.
(a) Set up an equation and find the ticket price.
(b) If the booking fee is doubled, how does that change the per-ticket effective price for 25 tickets?
(a) Set up an equation and find the ticket price.
(b) If the booking fee is doubled, how does that change the per-ticket effective price for 25 tickets?
Working space
8Problem 8 of 12
The man and his son. A man is currently four times as old as his son. In four years' time, he will be three times as old as his son will be then.
(a) Let the son's current age be . Write an equation modelling the situation.
(b) Solve and find the current ages.
(a) Let the son's current age be . Write an equation modelling the situation.
(b) Solve and find the current ages.
Working space
9Problem 9 of 12
Mixture problem. A container has 5 L of a mixture that is 20% acid. How many litres of pure water should be added to make a mixture that is 10% acid?
Working space
10Problem 10 of 12
Equation from a triangle. A triangle has sides , and cm. The perimeter is 36 cm.
(a) Set up and solve an equation.
(b) Find the three side lengths.
(c) Check the triangle inequality.
(a) Set up and solve an equation.
(b) Find the three side lengths.
(c) Check the triangle inequality.
Working space
11Problem 11 of 12
Solving for in a value-per-bag context. Tea is sold in two packs. A small pack contains tea bags for 3 chf; a large pack contains tea bags for 11 chf.
(a) Write expressions for the cost per tea bag for each pack.
(b) For which are the two packs equal value?
(c) For , which pack is better value?
(a) Write expressions for the cost per tea bag for each pack.
(b) For which are the two packs equal value?
(c) For , which pack is better value?
Working space
12Problem 12 of 12
Trial-and-improvement check. Show that the equation has a solution between 5 and 6, and use a bisection-style search to find to 1 d.p.
Working space