Joe buys 18 kg of potatoes. Some of these are old potatoes at 22¢ per kilogram, and the rest are new ones at 36¢ per kilogram.

(i) Denote the mass of old potatoes he buys by [tex] m [/tex] kg. Write down an expression for the total cost of Joe's potatoes.

(ii) Joe pays with a $5 note and receives 20¢ change. What mass of new potatoes does he buy?

Sure! Let's solve the problem step-by-step.

Joe buys a total of 18 kg of potatoes, consisting of both old and new potatoes. We will denote the mass of old potatoes as [tex]$ m $[/tex] kg. Therefore, the mass of new potatoes will be [tex]$ 18 - m $[/tex] kg.

### Step 1: Expression for Total Cost

1. Cost of Old Potatoes: Old potatoes cost 22 cents per kilogram. Hence, the cost for [tex]$ m $[/tex] kg of old potatoes is [tex]$ 22m $[/tex] cents.

2. Cost of New Potatoes: New potatoes cost 36 cents per kilogram. Therefore, the cost for [tex]$ 18 - m $[/tex] kg of new potatoes is [tex]$ 36 \times (18 - m) $[/tex] cents.

3. Total Cost: The total cost for all the potatoes is the sum of the costs of the old and new potatoes:
[tex]\[
\text{Total cost} = 22m + 36(18 - m)
\][/tex]

### Step 2: Calculate Mass of New Potatoes

Joe pays for the potatoes with a $5 note and receives 20 cents change. Therefore, the cost of the potatoes is:
[tex]\[
5 \, \text{dollars} - 0.20 \, \text{dollars} = 4.80 \, \text{dollars}, \text{ or } 480 \, \text{cents}
\][/tex]

We set up the equation for the total cost:
[tex]\[
22m + 36(18 - m) = 480
\][/tex]

Solving for [tex]$ m $[/tex]:

- Expand the equation:
[tex]\[
22m + 648 - 36m = 480
\][/tex]

- Combine like terms:
[tex]\[
-14m + 648 = 480
\][/tex]

- Subtract 648 from both sides:
[tex]\[
-14m = 480 - 648
\][/tex]
[tex]\[
-14m = -168
\][/tex]

- Solve for [tex]$ m $[/tex] (mass of old potatoes):
[tex]\[
m = \frac{-168}{-14} = 12
\][/tex]

Thus, Joe buys 12 kg of old potatoes.

- Calculate the mass of new potatoes:
[tex]\[
18 - m = 18 - 12 = 6
\][/tex]

Joe buys 6 kg of new potatoes.