Ex 2.02Scale Factor Method #2: Inverse ProportionIn the previous exercise, we learnt about DIRECT PROPORTION: That's when two quantities are related; as the first increases, the second also increases. That's applicable for situations like cost of sprouts and mass of sprouts (YUK!) But there are situations where as one quantity increases, the other decreases... An example is: the time it takes a ship to travel from from A to B and the speed of the ship. These are obviously related to each other, not in DIRECT PROPORTION, but in INVERSE PROPORTION, because: 1. If the speed was greater, the journey would take less time 2. Double the speed would halve the journey time When two quantities are in inverse proportion we can still use the scale factor method, but we need to ADAPT the method... e.g. A ship travelling at 40 knots will make the journey from Portsmouth to Calais in 168 minutes. How long would the journey take at 48 knots?Step 1: We know that at 40 knots it takes 168 minutes, so we write this first: ┌─┬─┬────────────────────┬──┬────────┤ These two vary in
▼ ▼ ▼ ▼ ▼ │ INVERSE proportion
SPEED TIME
--------------------------------
40 knots ► 168 minutes
Step 2: We want to find the time taken at 48 knots, so we write 48 knots under the 40 knots: SPEED TIME
--------------------------------
┌ 40 knots ► 168 minutes
Χ48 │
40 │
└►48 knots ► ___________
▲ ┌────────────────┐
└─────────────────────────────┤ Make sure the │
│ UNITS match up │
└────────────────┘
Step 3: ON THE L.H.S, the Scale-factor is: \(\times \frac { \color{#62be53}{48} }{ \color{#2b83c3}{40} }\) SPEED TIME
--------------------------------
┌ 40 knots ► 168 minutes ─┐┌────┐
Χ48 │ ││Χ40 │
40 │ ││ 48 │
└►48 knots ► __________◄──┘╘══╦═╛
╚════╡ For INVERSE PROPORTION
│ we FLIP the fraction
Step 4: So the answer is: \( 168\;\times\; \frac {\color{#2b83c3}{ 40 }}{\color{#62be53}{ 48 }} \;=\; \color{#bd398c}{140} \)minutes
Scale Factor Method #3: Indirect ProportionWe can also adapt the Scale-factor method for another, slightly harder situation: If, as the first quantity doubles, second quantity doesn't just double (perhaps it quadruples...) Expressed in that abstract way, that is hard to understand - but a situation makes it easy to understand: The Price of a TV and its Screen-size: A 20-inch TV might be priced at £100, whereas a 40-inch TV is priced at £400. It is probably much more realistic to say that the Price of a TV varies in proportion to the SQUARE of the Screen-size And we can easily adapt our method to deal with that: e.g. Panasonic make two identical OLED TVs, one with screen size 42-inches priced at £600 and another with screen-size 63-inches. How much should I expect to pay if price is proportional to square of screen size?Step 1: We know that 2.5 kg of apples costs £1.20, so we write this first: SIZE² PRICE
-----------------------------
42² inch² ► £600
└─┬─┘ ┌───────────────────────────────┐
└────────────────────────────────┤ Price depends on Screen-size² │
│ So, we must SQUARE the sizes │
└───────────────────────────────┘
Step 2: We want to find the cost of a 63 inch TV, so we write 63² inch² under the 42² inch²: SIZE² PRICE
-----------------------------
42² inch² ► £600
63² inch² ► ____
▲ ┌────────────────┐
└───────────────────────────┤ Make sure the │
│ UNITS match up │
└────────────────┘
Step 3: The Scale-factor is: \(\times \frac { {63}^{2} }{ {42}^{2} }\) SIZE² PRICE
-----------------------------
┌ 42² inch² ► £600
Χ63² │
42² │
└►63² inch² ► ____
Step 4: The SAME Scale-factor (\(\times \frac { {63}^{2} }{ {42}^{2} }\)) will also convert the £600: SIZE² PRICE
-----------------------------
┌ 42² inch² ► £600 ─┐
Χ63² │ │ Χ63²
42² │ │ 42²
└►63² inch² ► ____◄─┘
Step 5: So the answer is: \( £600\;\times\; \frac { \color{#62be53}{ {63}^{2} } }{ \color{#2b83c3}{ {42}^{2} } } \;=\; \color{#bd398c}{£1350} \)
In fact, we could combine indirect and inverse proportion to have a situation such as: The efficiency of a car varies inversely with the square of the speed at which the car is travelling... We'll encounter a couple a few of those questions, but later on, we'll also learn an algebraic method for dealing with those questions...
For each of these questions:
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This is an interesting question: It says, "if the stamps could be arranged in a block 24 cm wide..." Why the use of "if"??? Well - if we imagine this (on the left) is the block of stamps (it isn't necessarily): Then would it be possible to re-arrange this into a block that we 24 cm wide? Okay - so the answer to the 2nd part of the question is "NO, not necessarily"... But that doesn't stop us answering the 1st part...
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Question 13: More people means less time, so this is for "how long will it take", we can use INVERSE PROPORTION
As for, "what load will each carry": Again, More people means less weight each - so again, that's INVERSE PROPORTION
The only reason you might fail is that you don't know how to quickly-and-easily convert from hours, minutes and seconds to hours...
I showed you how to do this in Question 8, but I'll show you again:
REMEMBER, I'M NOT USING THE NUMBER YOU NEED TO CONVERT:
To convert: 2 hours, 25 minutes and 12 seconds into HOURS:
Press:
2
°"
25
°"
12
°"
=
Finally, press:
°"
once more
You have the answer in hours now!
To convert 2.42 hours back into hours, minutes and seconds:
Type:
2.42
=
Press:
°"
And you have it in hours, minutes and seconds...
Question 14: This is exactly the same question as Qu 3 and as Qu 4, but just in disguise...
If you break-it-down, it's just saying:
№ in Bunker Sufficient for:
---------------------------------
┌ 4 people ► 5 years ─┐
Χ
│ │
│ │
└►7 people ► ______◄──┘
(Is it inverse proportion?)
...maybe we should give our answer in __ years and __ days...
Question 15: There are two thing that are affected by the changing frame rate - so deal with each one separately...
We know that a higher frame rate uses more memory:
Frame- rate File-size:
---------------------------------
┌ 14 fps ► 112 Kb ─┐
Χ
│ │
│ │
Blurry:[A]---└►6 fps ► ______◄──┘
(Is it inverse proportion. or direct proportion?)
...then we need to work out the file size for [C]
...then we need to think about the blur levels for both [A] and [C]