Coaching Centre
Substitution and spreadsheets
Substituting into linear and non-linear formulas, using brackets on the calculator, rounding with units, and writing spreadsheet formulas with relative and absolute cell references.
A formula is a recipe. The letters are the ingredients, and substitution means putting the actual amounts in. Almost every topic in Standard maths uses one, so the goal here is to do it the same careful way every time, with no slips.
A routine that doesn't slip
Most lost marks in substitution questions come from rushing the calculator, not from the maths. Use the same four lines every time:
- Write the formula.
- Rewrite it with the numbers in place of the letters, putting each number in brackets if it is negative or a fraction.
- Evaluate it in one go on the calculator, or in clear steps.
- Round as asked and write the units.
Writing line 2 is not a waste of time. If the answer looks odd, line 2 shows the marker (and you) exactly where it went wrong, and you can still earn a working mark.
A function centre charges using the formula $C = 450 + 38n$, where $C$ is the cost in dollars and $n$ is the number of guests. Find the cost of a party for $62$ guests.
Powers, roots and negatives
A linear formula only adds and multiplies by numbers, like the one above. A non-linear formula has the variable squared, square rooted, on the bottom of a fraction, and so on. The routine is the same. The difference is that the calculator now needs more care.
Body mass index is $\text{BMI} = \dfrac{m}{h^2}$, where $m$ is mass in kilograms and $h$ is height in metres. Find the BMI of a person who is $68$ kg and $1.75$ m tall, to one decimal place.
The distance to the horizon is roughly $d = 3.57\sqrt{h}$, where $d$ is in kilometres and $h$ is your eye height in metres. How far can you see from the top of a $36$ m lifeguard tower on a clear day?
Formulas in a spreadsheet
A spreadsheet is a formula machine. Each cell has an address: column letter, then row number, so B2 is column B, row 2. A cell that starts with = holds a formula, and it can use other cells as its ingredients.
| Maths | Spreadsheet |
|---|---|
| $\times$ | * |
| $\div$ | / |
| $x^2$ | B2^2 |
| $\sqrt{x}$ | SQRT(B2) |
| total of B2 to B6 | SUM(B2:B6) |
The real power is filling down. Write one formula in row 2, drag it down, and the spreadsheet copies it to every row, shifting the row numbers as it goes: =B2*C2 becomes =B3*C3, then =B4*C4. That shifting is called a relative reference.
Sometimes you don't want a cell to shift. If one cell holds the hourly rate for everyone, every row must keep pointing at that one cell. Dollar signs lock it: =B2*$F$1 fills down as =B3*$F$1, =B4*$F$1, and so on. That is an absolute reference.
Pick the formula =B2*E1 and look at the pay column. Tap a pay cell to see the formula it actually holds. Why is everyone after the first person paid $\$0$? Now pick the locked version and check the total.
Your turn
Find the BMI of a person who is $54$ kg and $1.62$ m tall, to one decimal place.
Show the working
$\text{BMI} = \dfrac{54}{1.62^2} = \dfrac{54}{2.6244} = 20.57\ldots \approx 20.6$
B2$F$1Use $v = u + at$ to find $v$ when $u = 4$, $a = 2.5$ and $t = 6$.
$v = 4 + 2.5 \times 6 = 4 + 15 = 19$
Use $A = \pi r^2$ to find the area of a circle of radius $2.4$ m, to two decimal places.
$A = \pi \times 2.4^2 = \pi \times 5.76 = 18.0955\ldots \approx 18.10$ m$^2$
Find $y$ when $x = -4$ in $y = 3x^2 - 2$.
$y = 3 \times (-4)^2 - 2 = 3 \times 16 - 2 = 46$. The bracket makes the square positive.
Convert $35^\circ$C to Fahrenheit using $F = \dfrac{9C}{5} + 32$.
$F = \dfrac{9 \times 35}{5} + 32 = 63 + 32 = 95$
A car's kinetic energy is $E = \tfrac12 mv^2$ joules. Find $E$ when $m = 1500$ kg and $v = 20$ m/s.
$E = 0.5 \times 1500 \times 20^2 = 0.5 \times 1500 \times 400 = 300\,000$ J
Cell D2 holds
=B2*C2. It is filled down to D3. What formula is in D3?Both references are relative, so both row numbers go up by one.
Cell D2 holds
=B2*$F$1. What is in D5 after filling down?B2 is relative so it becomes B5 (three rows down).
$F$1is locked, so it doesn't change.Cell B2 contains $3$. What value does
=2*B2^2show?Powers come first: $3^2 = 9$, then $2 \times 9 = 18$.
Exam style: A school hires a bus using $C = 180 + 2.75k$, where $C$ is the cost in dollars and $k$ is the number of kilometres. Find the cost of a $96$ km excursion.
$C = 180 + 2.75 \times 96 = 180 + 264 = 444$, so it costs $\$444$.
Exam style: Use $d = 3.57\sqrt{h}$ to find how far, in km, you can see from a cliff $25$ m above the sea. Give your answer to two decimal places.
$d = 3.57 \times \sqrt{25} = 3.57 \times 5 = 17.85$ km