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# Rearranging Formulas – a couple of examples

In my post a couple of days ago, I wrote about how a formula can be changed to make another ‘part’ of it the subject. The subject is the element that stands alone and its easier to find a value if the value to be found is the subject

In this post I’ll give a couple of examples.

Most of the world now measures temperature on the Celsius scale, but in a few places, most noticeably perhaps the USA, how hot it is on a weather forecast will be given in degrees Fahrenheit .

There is a simple formula for turning a temperature on one scale into a temperature on the other.

F = 9C/5 + 32  – For example, the boiling point of water is 100 deg C.
F = 9 * 100/5 + 32  = 180 + 32 = 212  – and this is right, 212F is the boiling point of water on the Fahrenheit scale.

But what if we were in the USA, and seeing the weather forecast would like to know the temperature in the more familiar C scale.

We can rearrange the formula so this is  C = . To do this we follow steps familiar to you if you can solved equation. The rule that stays the same is you need to keep the balance – a change you make to one side of the = you make the same change to the other.

F = 9C/5 + 32

(Subtract 32 from both sides)

F – 32 = 9C/5

Multiply both sides by 5/9
C = 5(F-32)/9

You will see that I’ve also switched the sides round, and written the new right side using Brackets.  Its F-32 that needs to be multiplied by the factor 5/9 and we have to remember our BODMAS rules.

Lets try out the new formula.  If 77 degrees F is given as the temperature
C = (77-32) * 5/9  = 45 * 5/9 = 25C – which is a warm day by UK standards.

Now consider v = u + at – which is a formula of motion used at A-Level. Lets re-arrange this to make t the subject.

Take u from both sides (and switch)
at = v – u
t = (v – u)/a

# A few notes on using formulas

Using formulas is part of Maths which can be really useful in real life situations – and when you are using Maths is Science  – and scientists use Maths all the time!

For example  the formula for working out speed* is
s = d/t  where s is the speed, d the distance travelled and t the time taken

So if we know that a car travels 120km in two hours, the speed overall is  120/2 = 60km/h  (Even the unit for speed tells us how to work it out, which is actually how I usually remember it)

Another question we might be asked is – If a car travels at 40km/h for 30 minutes, how far will it travel?

Here we have the speed and the time, and we can plug our numbers into the formula

40 = ? / 0.5  – we can change this to be ? = 20km by multiplying both sides by 0.5 (30 minutes, hence half an hour)

Which is OK, but a bit awkward, especially if we have a lot of similar calculations

So…. the best thing is to re-arrange the formula to the quantity we always want to find is one the left hand side….  and this is where algebra comes in.

s = d/t.  We can rearrange this formula by multiplying both sides by t.  We may not know what t is but it has a value

s [* t] = d/t [* t]  –  so s * t = d

We reverse the formula, so the single letter is on the left.

d = s * t or d = st (because mathematician are sometimes lazy and leave out the multiply sign)

We call this the ‘subject’ of the formula. In fact, thats how you may be asked to do this in an exam – ‘Make s the subject of the forumla’

I will post with more examples tomorrow

*NOTE: As one of my students mentioned last week, we ‘should be using velocity now rather than speed’.  He had a point, but remember, velocity is the measure where direction is important too..  speed is a useful measure if the direction you are traveling is known or somehow less important.

# The Teacher’s response!

So..  what happened!     Kate and Jo had to sit the test on Tuesday.  After the test, Kate goes up to the teacher and says angrily ‘You didn’t play by your rules!’

“How do you mean?” asks the teacher, patiently.

So Kate explained to the teacher what she had explained to Jo on Friday.

“That’s very clever,”admitted the teacher. “But tell me, after working all that out, did you expect the test to be today until I told you that it would be?”

“Oh!”  said Jo.  Kate just scowls.

# The puzzle of the school test

On a Friday, a teacher say’s to Jo’s class – “Next week you will have a surprise test, but you will not know until you arrive at school that day, that the test is that day.”

Jo is very worried – she hates tests and she hates surprises.  But her friend Kate says – ‘Its OK, we won’t be having a test’”

“But the teacher said…”

“Just think about it,” suggests Kate.  “We can’t have the test on Friday, because if the test is on Friday, we would know it was on Friday when we go home on Thursday and we havn’t had the test yet. And that breaks the rule.”

“So it won’t be Friday,” says Jo. “It can be one of the other days.”

“Once we know it can’t be Fridyam,, it can’t be Thursday for the same reason!”

“I don’t get it.”

“We know it can’t be Friday, so when we leave school on Wednesday, the test must be Thursday.  Breaking the rule!”

“But if we know it can’t be Thursday or Friday,” says Jo, seeing what her friend is saying at last, “then it can’t be Wednesday.”

“Exactlty!  So we can’t have a test unless our teacher breaks the rules!”

So is Kate right? Has the teacher set himself an impossible rule?”