Online Math Center

SAT Math: Ratios, Rates, Proportions

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. 

At the Online Math Center…

You will find experienced and devoted math tutors who will tailor the learning experience according to your needs. We offer individual or group tutoring to middle school students, but also to students that want to participate in math competitions. Contact us in order to find out more about our SAT Math tutoring services!

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