Costing/inventory models | Year 11

01 / LEARN

Understand the idea.

Connect quantities and prices

An order total adds each quantity times its unit price. Put quantities in product columns and prices in a column with the same product order. Their matrix product gives one total per quantity row, measured in dollars.

Check one total by hand to confirm the labels and units. Keeping the quantity matrix separate from the price column lets you compare price changes without altering the order quantities.

Interpret assumptions

Closing stock follows the accounting rule opening stock plus deliveries minus sales. Align the categories and ensure all three matrices describe the same period. Value the resulting quantities only after calculating the stock balance.

The calculation is only as complete as its records and assumptions. Missing deliveries or duplicated sales can produce impossible counts. Costs may also exclude freight, tax or discounts; identify what the supplied prices include before making a recommendation.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Cost two orders

Notebook/pen quantities are $Q = (\begin{array}{cccccccccc} 3 & 2 \\ 1 & 4 \end{array})$ and prices $P = (\begin{array}{cccccccccc} 5 \\ 8 \end{array})$ dollars. Find costs.

Worked solution

1
Match products and dimensions.
$(2 \times 2) (2 \times 1) → 2 \times 1$
Both quantity columns and price rows list notebooks first, then pens.
2
Cost the first order.
$c_{1} = 3 (5) + 2 (8) = 15 + 16 = 31$
Three notebooks and two pens contribute their individual costs.
3
Cost the second order.
$c_{2} = 1 (5) + 4 (8) = 5 + 32 = 37$
The second row contains one notebook and four pens.
4
Present the cost column.
$Q P = (\begin{array}{cccccccccc} 31 \\ 37 \end{array}) dollars$
The two entries preserve the original order-row order.

Answer

$Q P = (\begin{array}{cccccccccc} 31 \\ 37 \end{array})$ dollars.

EXAMPLE 02

Closing stock

Opening $S = (\begin{array}{cccccccccc} 5 & 8 \\ 7 & 4 \end{array})$ , deliveries $D = (\begin{array}{cccccccccc} 3 & 2 \\ 0 & 5 \end{array})$ and sales $R = (\begin{array}{cccccccccc} 4 & 3 \\ 2 & 6 \end{array})$ . Find closing stock.

Worked solution

1
Write the stock balance.
$S_{closing} = S + D - R$
Deliveries increase available stock, while sales decrease it.
2
Combine opening stock and deliveries.
$S + D = (\begin{array}{cccccccccc} 5 + 3 & 8 + 2 \\ 7 + 0 & 4 + 5 \end{array}) = (\begin{array}{cccccccccc} 8 & 10 \\ 7 & 9 \end{array})$
Each delivery is added to its matching stock category.
3
Subtract the sales quantities.
$S_{closing} = (\begin{array}{cccccccccc} 8 - 4 & 10 - 3 \\ 7 - 2 & 9 - 6 \end{array}) = (\begin{array}{cccccccccc} 4 & 7 \\ 5 & 3 \end{array})$
Sales remove items from the same row-and-column positions.
4
Check physical feasibility.
Every closing count is non-negative, so no category records a sale beyond its opening stock plus deliveries.

Answer

$(\begin{array}{cccccccccc} 4 & 7 \\ 5 & 3 \end{array})$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Find the cost $(\begin{array}{cccccccccc} 4 & 3 \end{array}) (\begin{array}{cccccccccc} 6 \\ 9 \end{array})$ dollars.

Show answer & reasoning

51 dollars.

Take the row-column total: $4 (6) + 3 (9) = 24 + 27 = 51$ dollars. Each quantity is multiplied by its corresponding unit price.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Find order costs for $Q = (\begin{array}{cccccccccc} 2 & 5 \\ 4 & 1 \end{array})$ and $P = (\begin{array}{cccccccccc} 7 \\ 3 \end{array})$ .
2 marks
Stock is $(\begin{array}{cccccccccc} 20 \\ 15 \\ 8 \end{array})$ and sales $(\begin{array}{cccccccccc} 7 \\ 4 \\ 3 \end{array})$ . Find remaining stock.
2 marks
A quantity matrix has 3 shop rows and 4 product columns. What order must a price column have?
2 marks
Three items priced $p$ dollars each and two items priced 4 dollars each cost 29 dollars. Find $p$ .
2 marks
A stock calculation returns $- 2$ items. Give a sensible response.
2 marks
A subtotal of 80 dollars attracts a uniform 10% delivery surcharge. Find the total.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find order costs for $Q = (\begin{array}{cccccccccc} 2 & 5 \\ 4 & 1 \end{array})$ and $P = (\begin{array}{cccccccccc} 7 \\ 3 \end{array})$ .
2 marks
$2 (7) + 5 (3) = 29$ .
$4 (7) + 1 (3) = 31$ .
$(\begin{array}{cccccccccc} 29 \\ 31 \end{array})$ dollars.
Stock is $(\begin{array}{cccccccccc} 20 \\ 15 \\ 8 \end{array})$ and sales $(\begin{array}{cccccccccc} 7 \\ 4 \\ 3 \end{array})$ . Find remaining stock.
2 marks
Subtract each sold count.
$20 - 7 = 13$ , $15 - 4 = 11$ , $8 - 3 = 5$ .
$(\begin{array}{cccccccccc} 13 \\ 11 \\ 5 \end{array})$ .
A quantity matrix has 3 shop rows and 4 product columns. What order must a price column have?
2 marks
Four rows match the products.
One column produces one cost per shop.
$4 \times 1$ .
Three items priced $p$ dollars each and two items priced 4 dollars each cost 29 dollars. Find $p$ .
2 marks
$3 p + 8 = 29$ .
$p = \frac{21}{3} = 7$ .
7 dollars.
A stock calculation returns $- 2$ items. Give a sensible response.
2 marks
A negative count is impossible.
Check deliveries, sales and subtraction order.
Reconcile the records before reporting stock.
A subtotal of 80 dollars attracts a uniform 10% delivery surcharge. Find the total.
2 marks
The surcharge is $0.10 (80) = 8$ dollars.
Add it once: $80 + 8 = 88$ .
88 dollars.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: matrices and networks. Original learning resources follow the approved Ascent Library chapter map.

Official curriculum source ↗

In this lesson

Understand the idea.

Connect quantities and prices

Interpret assumptions

See the method in action.

Cost two orders

Match products and dimensions.

Cost the first order.

Cost the second order.

Present the cost column.

Closing stock

Write the stock balance.

Combine opening stock and deliveries.

Subtract the sales quantities.

Check physical feasibility.

Try it yourself.

Check your understanding

Keep practising