04 Solving third type of percent problem: Finding percentage

Finding percentage

The previous types of percent problems we dealt with were of forms:

P% of A = ?

P% of ? = B

This third type is to find out what the percentage itself is. For example, say 10 out of 50 students are absent in a class. What percentage of students are absent? This is asking us what percent is 10 out of 50.

So how do we find this? Let’s start out with a very basic example first. Say 60 out of 100 people in the shopping mall are teenagers. What percentage of people are teenagers? Well, we know that the ratio of teenagers to total people can be written as 60 : 100, or in fractional form of 60/100. So, 60 out of every 100 people are teenagers. This can be written in terms of percentage -> 60%, since we know that anything per 100 can be written as percent.

Going back to our absent students example, we see that the ratio of absentees to total students is 10 : 50, or in fractional form 10/50. Here, the absentees are not in the form of per 100. But using the idea of equivalent fractions, we can write 10 : 50 and 20 : 100 (multiplying both by 2). This tells us that 20% of students were absent. Quite high! This step was easy since we could easily change 50 to 100, but what if there are cases where the whole cannot be easily converted to 100?

Let’s try to find percentages using the ratio method, and take help from tables and double number line for such cases.

Out of the 90 kg of food purchased by a restaurant, 36 kg was all meat products. What percentage of the food was meat products?

To solve this, let’s first form a ratio of meat products to overall food -> 36 : 90. What we need to find is the meat per 100 parts of food. To do this, we need to first find the meat per 1 part food. Then we can find the meat per 100 parts of total food. Let’s use a table to do this.

Here, we see that in order to find the meat per 100 parts of total food, which is simply our percentage, we could simply multiply the fraction we get (36/90) by 100. This gives us the answer as 40. The ratio is 40 : 100, and the percentage is 40%.

The same could be applied for the 10 : 50 case from before. We multiply the fraction 10/50 by 100, giving us 20%!

There is another way to think of finding percentages using the table and the ratio method. Instead of finding the ratio of meat to total food, we can form a ratio for total food and the percentage it represents, which is 100%. The ratio is 90 : 100.

Now, we need to find the percent for 36 kg food, which we can find by first finding the percentage of 1 kg food and then multiplying by 36.

In this case as well, we see that the final percent is found by multiplying the fraction of meat product out of total by 100.

Lastly, let’s confirm the answer using the double number line diagram.

If 90 is 100%, they align on the two lines. We can divide 90 into 10 parts to show each part measuring 9 kg. This gives us 36 kg at the fourth mark after 0. Similarly, we can divide 100% into 10 parts, with each part measuring 10%. 36 kg now aligns with 40%!

If we go back to our two other types of percent problems, we can compare all three now:

  1. P% of A = ?
  2. P% of ? = B
  3. ?% of A = B

We see that it is essentially the same problem with different missing values. Once we know how to form the equation, solving it is not difficult. Like we did for types 1 and 2, we can also easily solve the third type numerically, without using number lines. We have already seen that the percentage can be found by multiplying the fraction by 100. If we look at the third equation, we have ?% of A = B

?/100 x A = B

? = B/A x 100

We get the same thing using the equation, obviously! Since the answer is in percentage, we can write it as B/A x 100%.

To conclude, if we need to find what percentage of a number B is of another number A, we simply find the fraction of B/A and then multiply this by 100%. This gives us the formula as B/A x 100%. Think of it as converting a fraction into a percentage. They both mean the same thing, they’re simply different representations. This also makes it easy for us to think of decimals in terms of percentages, if needed. If 0.25 people are absent in a class, we should easily be able to tell that 0.25 x 100% people are absent -> 25%! We are simply converting the decimal into a percentage!

A small interesting note: The reason multiplying by 100% works is because 100% simply means 100/100 as well. So, it is like multiplying a fraction by 1, which essentially gives the same value! Additionally, say you wrote B/A x 100% as B/A x 100/100. We could write it as (100B/A)/100, which gives us the percentage as 100B/A % OR B/Ax100 %! The same answer again!