Modelling with systems of equations Problem set B

Three students buy supplies at a school shop. Zoe buys 4 pens, 3 notebooks and 2 rulers and pays $13.90. Yusuf buys 2 pens, 5 notebooks and 1 ruler and pays $15.35. Xan buys 3 pens, 1 notebook and 6 rulers and pays $11.25. Find the cost of one ruler.

Solve the system and find all possible values of \(x\), \(y\) and \(z\): \(x^2 + 2y^2 + z^2=42\), \(3x^2 - y^2 + z^2=80\), \(x^2 + y^2 - 2z^2=11\).

A chemist mixes three salt solutions to make 600 ml of a mixture. Solution X is 10% salt, solution Y is 25% salt and solution Z is 40% salt. There is 100 ml more of solution Y than of solution X. The mixture contains 180 ml of salt. How much of solution Z is used?

A juice bar makes three kinds of smoothie. A classic uses 120 g of mango and 150 g of yoghurt. A tropical uses 80 g of mango and 60 g of yoghurt. A green uses 60 g of mango and 40 g of yoghurt. The classics sell for $4.50 each, the tropicals for $3.80 each and the greens for $5.20 each. In one day all the ingredients used are 6.6 kg of mango and 6.66 kg of yoghurt, and the total sales are $319.80. How many tropicals are made?

One watermelon has the same mass as 5 pineapples, and the same mass as 20 kiwi fruit. 3 watermelons and 4 pineapples have a mass 5 520 g more than 30 kiwi fruit. Find the total mass of one watermelon, one pineapple and one kiwi fruit.

A three-digit number has digits \(a\), \(b\) and \(c\), in that order, so the number is \(100a+10b+c\). The digits add up to 17. The middle digit is 2 more than the last digit. When the digits are reversed, the number decreases by 594. Find the number.

Three tables order breakfast at a bakery. Table 1 buys 3 croissants, 2 buns and 2 pastries and pays $15.50. Table 2 buys 1 croissant, 4 buns and 3 pastries and pays $16.90. Table 3 buys 5 croissants, 1 bun and 1 pastry and pays $15.45. Find the total cost of one croissant, one bun and one pastry.

A school club survey records the activity chosen by students. The pie chart shows how students chose between music, drama and chess. The sector for music has angle 90°. There are 4 more in drama than in chess. The number in music and chess together is 6 more than the number in drama. One person is chosen at random. Find the probability that they chose drama.

Three numbers \(a\), \(b\), \(c\) are in order, \(a\le b\le c\). The mean equals the median. The range is 3 more than the median. The difference between the two largest numbers is 5. Find the largest of the three numbers.