L1 Mathematical Reasoning
Question Bank
The graph shows the lines $2x + y = 11$ and $x - y = 1$. The lines cross at the point Q.
Find the coordinates of Q using an algebraic method. Clearly show all steps of your working.
The image below shows the total cost for different combinations of jeans and shirts.
By forming and solving simultaneous equations, find the total cost of one pair of jeans and one shirt.
In a farmyard containing only chickens and sheep, there are a total of 20 heads and 56 legs. How many chickens and how many sheep are in the yard?
Phone Company A offers a plan for \$30 per month plus \$10 for every gigabyte of data used. Company B offers a plan for \$50 per month plus \$5 for every gigabyte of data used.
By forming two linear equations, calculate the number of gigabytes of data for which the monthly cost of both plans would be identical.
'Zoom Rentals' charges \$70 per day and a one-time administration fee of \$50 to rent a car. 'Coastline Cars' charges \$85 per day with no administration fee.
After how many days would the total rental cost from both companies be exactly the same?
A company offers two different weekly payment structures for its sales team.
Option 1: A base salary of $400 per week, plus a 10% commission on all sales.
Option 2: A base salary of $700 per week, plus a 5% commission on all sales.
What amount of weekly sales would result in the same total earnings from both options?
A mobile mechanic charges a \$55 call-out fee and $80 per hour for labour. A workshop charges a flat rate of \$110 per hour with no call-out fee.
By forming two linear equations, determine the number of hours for which the total cost from both providers would be the same.
(i) Solve the simultaneous equations
$3x + 2y = 16$$5x - 3y = 14$
(ii) The line $kx - y = 6$ passes through the same point. Find the value of $k$.
Line C has the equation $3x+2y=7$. Line D has the equation $2x+3y=3$. Find the coordinates of the point where the two lines intersect.
The image below shows two meals and their total calorie counts. The first meal, consisting of 3 slices of pizza and 2 hot dogs, has 1760 calories. The second meal, with 2 slices of pizza and 1 hot dog, has 995 calories.
By forming and solving simultaneous equations, find the number of calories in a single slice of pizza and in a single hot dog.
Line A has the equation $y = 4x - 5$. Line B has the equation $3x + 2y = 22$. Find the coordinates of the point where the two lines intersect.
Sarah has \$60 more than Tom. After they each spend \$10, Sarah finds that she has three times as much money as Tom.
By forming and solving simultaneous equations, calculate how much money each person had at the start.
The average age of a father and his son is 25. Half the difference in their ages is 15.
By forming and solving simultaneous equations, find their current ages.
Two wizards, Alistair and Beatrice, are comparing their supply of mana potions.
Alistair says to Beatrice, "If you give me 10 of your potions, the ratio of my potions to yours will be 2 to 1."
Beatrice replies, "If you give me 10 of your potions instead, the ratio of my potions to yours will be 2 to 1."
By forming and solving simultaneous equations, find out how many mana potions each wizard has.
Two friends, Alex and Ben, are comparing their collections of magic crystals in a video game.
Alex says to Ben, "If you give me 5 of your crystals, we'll have the same number."
Ben replies, "If you give me 5 of your crystals instead, I'll have twice as many as you!"
By forming and solving simultaneous equations, find out how many crystals each player has.
Two numbers, $a$ and $b$, are in the ratio $a:b = 5:2$ and have a sum of $a+b=35$.
Find the values of $a$ and $b$, and use them to simplify the expression below:
$ax^b - bx^a + 5ax - 2bx$
The total cost of 3 adult tickets and 2 child tickets to a movie is \$77. The cost of 5 adult tickets and 3 child tickets is \$124. What is the individual cost of an adult ticket and a child ticket?
A mother is currently four times as old as her son. In 4 years' time, she will be three times as old as her son will be. How old are they now?
The diagrams show a cat and a table. When the cat stands on top of the table, the total height from the floor to the top of the cat's head is 137 cm. When the cat is on the floor, the distance from the top of its head to the tabletop is 91 cm.
By defining two variables and using an algebraic method, find the height of the table and the height of the cat.
The diagram shows a shape made from five congruent rectangles. The total width of the shape is 37 cm and the total height is 26 cm. By forming and solving simultaneous equations, find the area of the shape.
A rectangular garden has a perimeter of 34 metres and an area of 60 m². What are the dimensions of the garden?
The diagram below shows two straight lines. Using an algebraic method, find the coordinates of the point where the two lines intersect. You must show how you derived the equations of the lines.
