Ratios
Ratios basically tell you the relationship between two or more quantities.
Seems difficult, but it's not, and it's essential in lots of things, from Maths to Science to cooking to money.
The Basics One example is a Grandma's recipe for Pop Rocks (fizzy candy): 2 spoons of gunpowder to 1 spoon of sugar or
Gunpowder Sugar
2 : 1
In other words, ratios tell you how much something is compared to something else.
For example, in a class of 45 child, the ratio of boys to girls is 5:4. So for every 5 boys, there are 4 girls. This is also used in photos. The height:width ratio of a photo is 2:3. Which means if the flag is 2m tall, it will have a width of 3m, and a height of 2cm will have a width of 3cm.
We can also express ratios as a fraction:
Commonly used in things like speed, examples are like
5m/s
Which means for every 1 second a car will move 5 meters. Scaling Ratios
You can scale a ratio up or down, that is make it bigger or smaller but their relative size are the same. Whaaaat? What is that?
Before we dive into the confusing stuff, let's look back at Grandma's Pop Rocks:
With 2 spoons of gunpowder and 1 spoon of sugar, we can make 1 packet of Pop Rock.
What if our friends are coming and we wanna make 10? Easy, right? Just multiply all ingredients by ten! 2:1 Multiply both quantities by 10
20:10 and we will get 10 packs of Pop Rocks! That's basically how scaling works. You multiply all quantities with a common factor - the same number.
Some people call it a scalar quantity, where you multiply the same number with all the values. In the end, as you can see, you'll still get the same thing, just in higher quantities or lower quantities or larger or smaller or...well it depends on the situation but you get my point, they'll be relatively the same.
Scaling does not have to be a number - it can be the units. If both units are the same, then the ratios are proportional. 2 cm : 5cm is proportional to 2 m : 5 m.
And you can also do this with maps and photos - it's called a scale. In maps roads can thus become smaller but still let you calculate the true distance, and in pictures you can print a large copy out, of the same relative size, even if the screen of your camera is tiny.
For example, if you want to magnify your Uncle-in-Mickey-Mouse-costume portrait, simply measure its dimensions, multiply it by a common factor, and you can buy a canvas of that size and repaint the portrait. In Maths, we call this Proportional, or Equivalent Ratios - they express the same meaning. We can simplify this by finding it's HCF and dividing all quantities by this.
Let's try an exercise: Grandma is a great cook but is bad at Maths. In her cookbook, she has multiply recipes that will produce different quantities of different snacks...can you help her simplify all the ratios so they will just make one unit of snack only?
1) Lobster Soup: (Lobster:Soup Base) 50:100 2) Pork Rolls: (Pork:Pastries) 4:4
3) Lemonade: (Lemons:Water:Syrup) 32: 80: 48
Answers:
1) 1:2 2) 1:1 3) 2:5:3
Unit Ratios
Ratios with 1 as one of its value. For example, 1:c or d:1. Good examples are stuff like speed and density. To calculate it, just divide one side by the other quantity. A:B
Divide A by B or B by A. A/B:1 or 1:B/A Comparing Ratios
Unit ratios are good for comparing things. For example, if you want to know which brand of drink has the highest amount of sugar...
Water:Sugar
Brand 1: 10:5 -> 2:1
Brand 2: 80:20 -> 4:1
Brand 3: 32:4 -> 8:1
Wow! Brand 3 obviously has the highest amount of sugar due to its ratio! Definitely get that! Great for kids who wanna be hyperactive!
(These are nice numbers, but most of the times you will get fractions. Such life, right?)
Cross Product
In exams, you may get questions like thi:
If A:B is 2:3
and B:C is 5:7
then what is A:B:C? There is a method for this:
Line them up like this:
A:B
* B:C
_____
And multiple each value, first diagonal, then vertical, finally diagonal. Or A*B, then B*B, then B*C. Easier done that said, really! In our example...
A:B:C
2:3 5:7
____
10:15:21
There we go! Ta-da!
Finding Unknowns
Let's take Grandma's lemonade recipe:
2:5:3
And someone asks you how much lemon and water do we need if we have 60 pints of syrup? Let's write out everything first:
2:5:3 = L:W:60
There are a couple ways to solve this:
1) Find the common factor
Just do 60/3 and you will get the factor as you know these two ratios are proportional. What's the factor? 20! So multiply each by 20 and you will get the ratio.
2) Cross Multiplication Useful when quantities come in two.
One such example is
1/2 = 50/?
Here you multiple each by its opposites i.e. the numerator on the left by the denominator on the right. 1(?)=50(2)
100=1(?)
?=100
3) Sum of all parts
This is a great method personally and I use it all the time!
First, you have to understand the concept of parts.
Parts are basically adding everything up together.
2:5:3 -> 2+5+3=9 So there are 9 parts in a lemonade.
And it makes sense! You're putting 9 chunks of stuff to make a lemonade, right?
So now you know 3 parts of the drink is syrup.
So let's go back to that mystery ratio:
You know 60 pints constitutes 3 parts of the total amount of drink - in other words, 1 part of the drink is just 60/3=20 pints. So multiply that by 9 (as there are 9 parts) and you will know there will be 180 pints of lemonade produced. Of course, that's not what we are looking for. But if just multiply the parts by the amount 1 part has in our example, we will get the answer.
2(20):5(20):3(20)
Answer:
40:100:60
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Ratios are actually very easy, but becomes very important! (Cough * Chemistry * Cough)
