A corner shop weighs bags of flour on a balance. Every bag has the same mass of \(x\) kg. The shopkeeper puts 3 bags and a 2 kg weight on the left pan. She ...

A balance is a picture of an equation: whatever is on the left pan weighs the same as whatever is on the right pan. This question uses that idea to form a linear equation, solve it, use the result in a new balancing situation, and finish with an inequality.

(a) Equation from the balance. [2]
The left pan carries 3 bags and a 2 kg weight, so its total mass is \(3x + 2\) kg. The right pan carries 1 bag and an 8 kg weight, so its total mass is \(x + 8\) kg. Because the pans balance, the two totals are equal:

One mark for each side written correctly in terms of \(x\).

(b) Mass of one bag. [2]
Remove one bag from each pan (subtract \(x\)) and remove 2 kg from each pan:

Each bag has a mass of 3 kg. Check on the balance: the left pan holds \(3 \times 3 + 2 = 11\) kg and the right pan holds \(3 + 8 = 11\) kg, so it does balance. Doing exactly the same thing to both pans is what keeps the balance level, and that is why the same operation must be applied to both sides of an equation.

(c) Value of \(w\). [2]
Now the left pan holds 5 bags, mass \(5 \times 3 = 15\) kg. The right pan holds 1 bag plus weights totalling \(w\) kg, mass \(3 + w\) kg. Balancing:

So 12 kg of weights are needed. In effect the four extra bags on the left must be matched, and \(4 \times 3 = 12\) kg confirms it.

(d) Least number of bags with total mass at least 20 kg. [2]
If there are \(n\) bags, the total mass is \(3n\) kg, and "at least 20 kg" means

Bags come whole, and the number must be at least 6.66..., so \(n = 7\). Checking: 6 bags weigh 18 kg, which is under 20 kg, while 7 bags weigh 21 kg, which meets the condition. Marks: one for the inequality, one for rounding up to 7.

Examination point: the rounding direction depends on the inequality, not on the decimal. Here 6.66... rounds up to 7 because the total must reach 20 kg, even though ordinary rounding of a "just over 6" value might tempt you to write 6.