No questions found
This investigation is about the results of two calculations when you change the order of operations.
Each calculation uses three integers, $x$, $y$ and $z$.
Calculation 1:
Add $y$ and $z$. Multiply the answer by $x$.
Calculation 2:
Multiply $y$ and $z$. Add the answer to $x$.
Example
$x = 3$ $y = 5$ $z = 7$
Calculation 1
$y + z = 5 + 7 = 12$
$12 \times x = 12 \times 3 = 36$
Result = 36
Calculation 2
$y \times z = 5 \times 7 = 35$
$35 + x = 35 + 3 = 38$
Result = 38
Complete the table.
| $x$ | $y$ | $z$ | Result of calculation 1 | Result of calculation 2 |
|----|----|----|----------------------------|----------------------------|
| 4 | 7 | 5 | 48 | 39 |
| 4 | 7 | 6 | | 46 |
| 4 | 7 | 7 | 56 | |
| 4 | 7 | 8 | | |
| 5 | 9 | 7 | 80 | 68 |
| 5 | 9 | 8 | 85 | |
| 5 | 9 | 9 | | |
| 5 | 9 | 10 | | 86 |
520
516
534
Show that when $x = 3$, $y = 4$ and $z = 9$ the two calculations give the same result.
584
523
558
$y = 5$ and $z = 6$.
Find the value of $x$ when the two calculations give the same result.
526
595
589
(a) Complete the table.
$$\begin{array}{|c|c|c|c|c|}\hline
x & y & z & \text{Result of calculation 1} & \text{Result of calculation 2} \\\hline
6 & 11 & 12 & 138 & 138 \\\hline
7 & 13 & 14 & 189 & \\\hline
8 & 15 & 16 & & 248 \\\hline
9 & 17 & 18 & & \\\hline
\end{array}$$
[2]
(b) Use the table in \textit{part (a)} to write $y$ and $z$ in terms of $x$.
$y = \text{.............................................}$
$z = \text{.............................................}$ [2]
(c) The results of calculation 1 and calculation 2 are always the same.
Use your expressions for $y$ and $z$ in \textit{part (b)} to show that this statement is true.
[3]
563
582
542
In this question $x$, $y$ and $z$ are consecutive, for example 15, 16 and 17.
(a) Write $y$ and $z$ in terms of $x$.
$$y = \text{.....................}$$
$$z = \text{.....................}$$ [1]
(b) The results of calculation 1 and calculation 2 are the same.
Show that $x^2 - x - 2 = 0$. [4]
(c) The results of calculation 1 and calculation 2 are the same. $x$, $y$ and $z$ are consecutive.
Find two sets of values for $x$, $y$ and $z$. [3]
590
541
571
When the result of calculation 1 is the same as the result of calculation 2,
$$x(y + z) = x + yz.$$
\(\text{(a)}\) \(x(x - 1) = (y - x)(z - x)\)
Expand the brackets and rearrange to show that this becomes \(x(y + z) = x + yz.\) [2]
\(\text{(b)}\) The results of calculation 1 and calculation 2 are the same. \(x = 6\) and \(z > y.\)
Find all the possible values for \(x, y\) and \(z.\) [4]
574
600
587
A farmer uses 600 metres of fencing to make a rectangular field. The length of the field is $x$ metres.
(a) The area of the field is $A$ square metres.
Show that a model for $A$, in terms of $x$, is $A = x(300-x)$. [2]
(b) Find the range of possible lengths of the field.
.................................................. [1]
(c) Sketch the graph of the model $A = x(300-x)$.
[Image Placeholder]
[3]
(d) Find the maximum area of the field and the length which gives this maximum area.
Area = ..................................................
Length = ..................................................
[2]
569
506
592
The farmer uses a wall as one side of the field.
The farmer uses 500 metres of fencing for this field.
(a) The length of the field is $x$ metres. Find a model for the area of the field, $B \text{ m}^2$.
(b) Which model gives the greater maximum area of the field, the model for $A$ or the model for $B$? Find the difference between these two maximum areas.
Model ................................................. Difference ................................................ [3]
536
502
571
The farmer uses the model $A = x(300 - x)$.
These are the costs for making the field:
- grass seed $0.27 per square metre
- fertiliser $0.08 per square metre
- a total of $7000 for fencing and labour.
(a) The cost to make a rectangular field with length $x$ metres is $C$.
Show that the model for $C$ is $C = 105x - 0.35x^2 + 7000$.
(b) Use your answer to Question 7(d) to find the cost of making the field with maximum area.
(c) The farmer keeps sheep in a field.
Each sheep has an area of 450 square metres.
The farmer is given $5 for each sheep.
He uses the money that he receives for the sheep to reduce his costs for making the field.
(i) Use this information to change the model in part (a).
(ii) Find the cost for the field with the maximum area.
584
530
553
