Students participating in SAT math competitions will have to solve ratios, rates, and proportions problems. If you take a look at any SAT math practice test, you will see that ratios, rates, and proportions are among the most common challenges. So, let’s get started!
What are ratios and how do we solve them?
Definition
Ratios are in fact fractions, that compare between two numbers. They are called “ratios” because they have been initially written like 1:2 instead of 12. The reading of 1:2 would be one half is akin to a one to two ratio; 25 would be written like 2:5, and read like 2 fifths, or 2 to 5 ratio.
Solution
We have the following exercise: What is the ratio of a:b, where 3a=7b?
The first operation to do is turn it into a fraction, so we have ab; in order to do so we divide everything to 3, thus: 3a3= 7a3, resulting in a = 7b3.
The next step is to bring a over b, so we divide both sides to b, as:
ab = 7b3b , we eliminate the b’s and end up with ab = 73 , which leads us to the beautiful result of the ratio of a:b is 7:3.
What are rates and how do we solve them?
Definition
Rates are represented by the next formula: d = rt, where d is distance, r is rate, and t is time, the speed formula.
Solution
We will stumble at SAT math tests upon exercises like:
Mike can drive two laps at every 1.5 minutes in the national circuit. At this rate, how many minutes does Mike need to drive 12 laps? Laps refer to the distance of the circuit and 1.5 times is the time.
Our formula for this particular exercise is:
2 = r x 1.5
How do we find r? We divide the whole formula by 1.5, and:
21.5 = r(1.5)1.5 , we reduce 1.5 from the r side and end up with:
21.5 = r . Still, we want to simplify the fraction more, so we will multiply the fraction with 2, resulting in:
43= r.
We still need to find the time it takes Mike to drive those 12 laps in a 43 rate, so:
12 (laps) = 43 (time). We reduce the formula by multiplying with 34, thus
3412 = 43(t) x 34; we reduce the 34 from one side, and result in:
34 x 12 = t, which reduced will end up being 9. So the answer to the question is that it will take Mike 9 minutes to drive 12 laps.
What are proportions and how do we solve them?
The interesting thing about the above SAT math exercise is that it could be solved through proportions, like this:
21.5 = 12x, where x is the time. We multiply them in a cross, and obtain: 1.5 x 12 = 18 and 2 x.
18 = 2x, so x = 9.
Definition
Proportions are expressed through the next formula: ab = cd , so a fraction equals another fraction.
Solution
When we solve proportions, we always cross multiply the values in the fraction, so we will multiply a to d, and b to c, meaning that bc = ad.
Let’s solve a proportion!
We have 16= x12 . What’s next? Cross multiplication.
6x = 12, thus x = 2. The solution is 16= 212
Once you start to get familiar with solving these SAT math problems, you will find that rates, ratios, and proportions are quite fun to solve. Also, you can easily apply them in real life to find out the right answer in recipes, or construction projects.
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