Ratio and Proportion Flashcards
shows how the quantity of an object is related to the quantity of another object
ratio
Example: Aling Bela has 4 chickens and 8 pigs on her small farm. What is the ratio of her chickens to her pigs?
Solution: We can express the ratio of chickens to pigs that Aling Bela owns as 4 : 8
Example 1: For every 4 burgers you will buy, you have to pay PHP 128. What is the ratio of the number of burgers bought to the price you have to pay? Express the ratio in fractional form.
Solution: We can express the ratio of the number of burgers bought to the price you have to pay as 4 : 128. In fractional form, we can write this as 4⁄128.
Example 2: There are 15 science teachers in a public high school. In that same high school, there are 10 math teachers. What is the ratio of science teachers to math teachers in that public high school? Express the ratio in fractional form.
Solution: We can express the ratio of science teachers to math teachers in that public high school as 15 : 10. In fractional form, we write it as 15⁄10.
Using Ratio to Compare a Part to a Whole
We have already defined what ratios are. However, the ratios that we have tackled in our previous sections pertain to a comparison of a quantity of an object to the quantity of a different object.
This time, let us use the ratio to compare a part of a whole to the whole itself.
Suppose that you and your friends bought a pizza and sliced it into 8 equal parts. Suppose that you’re able to take 2 slices from it. What is the ratio of the slices of pizza you have (a portion of the whole pizza) to the total number of slices (the whole pizza)?
The given situation above might ring a bell to you. Yes, we can use fractions to show that comparison. In particular, fraction 2⁄8 can be expressed into a ratio as 2 : 8
Example: In a classroom, 15 students are male while 20 students are female. What is the ratio of female students to the total number of students in the classroom?
Solution: There are 20 female students in the classroom. Meanwhile, the total number of students in the classroom is the sum of the number of male students and the number of female students. In total, there are 15 + 20 = 35 students in that classroom.
Therefore, the ratio of female students to the total number of students in that classroom can be expressed as 20 : 35
indicates that the two ratios are equal. In other words, proportions are equivalent ratios.
Proportion
Two ratios are proportional if they are
equal
Example: Give a ratio that is equivalent or proportional to 2 : 9
Solution: We can determine a ratio equivalent or proportional with 2 : 9 by multiplying each number in 2 : 9 by the same number.
Let us try to multiply the numbers in 2 : 9 by 5.
(2 x 5) : (9 x 5) = 10 : 45
Hence, 2 : 9 = 10 : 45.
Note: The number that you can use to find a ratio that is proportional to 2 : 9 is arbitrary. If we multiply the numbers in 2 : 9 by the same number, we will come up with a ratio that is proportional to 2 : 9. In this example, I just arbitrarily used 5. You may use any number and multiply it to the numbers in 2 : 9 and you will come up with a ratio that is proportional to it. For example, I can multiply the numbers of 2 : 9 by 7 and obtain 14 : 63. 14 : 63 is also proportional with 2 : 9
Parts of a Proportion:
extremes and means
Here are the properties of proportion:
- The product of the means is equal to the product of the extremes
For every proportion a : b = c : d, then a x d = b x c
This property tells us that if we multiply the means and also multiply the extremes of a proportion, we will obtain the same number.
For example, suppose the proportion 4 : 3 = 12 : 9.
If we multiply the means: 3 x 12 = 36
If we multiply the extremes: 4 x 9 = 36
ratio and proportion 7
Note that the products of the means and the extremes are both equal to 36.
- The reciprocals of the ratios in a proportion are equal
Recall that the reciprocal of a fraction is its multiplicative inverse, or simply the same fraction but with the positions of the numerator and the denominator reversed.
For example, the reciprocal of 2⁄5 is 5⁄2.
Given a proportion, say a : b = c : d, we can express it in fractional form as a⁄b = c⁄d
If we get the reciprocal of both fractions in a⁄b = c⁄d, we have:
b⁄a = d⁄c
We can express b⁄a = d⁄c in ratio as b : a = d : c
This property states that if we take the reciprocal of each ratio in a proportion, the ratios are still proportional. In symbols:
a : b = c : d → b : a = d : c
- Switching the means or the extremes in a proportion will result in a proportion
Suppose the proportion 1 : 7 = 3 : 21. If we try to switch the positions of the means of this proportion, we have 1 : 3 = 7 : 21. You can verify using cross-multiplication that 1 : 3 = 7 : 21 is true (that is, 1 : 3 and 7 : 21 are equivalent ratios or proportional).
Now, let us try switching the extremes of 1 : 7 = 3 : 21. That is, we obtain 21 : 7 = 3 : 1. Again, you can verify using cross-multiplication that 21 : 7 = 3 : 1 is true.
Hence, for every proportion a : b = c : d, switching the means or the extremes will still result in a proportion.
a : b = c : d → a : c = b : d and d : b = c : a
Example 1: What must be N so that N : 8 = 2 : 16 is a proportion?
Solution: Let us use the fact that the product of the means of a proportion is equal to the product of the extremes.
Multiplying the means, we have: 8 x 2 = 16
Multiplying the extremes, we have 16 x N
Now, by the first property, 16 x N = 16. What must be multiplied by 16 so that it will be 16? That number should be 1.
Hence, N = 1.
Therefore, the proportion should be 1 : 8 = 2 : 16.
Example 2: Four kilos of chicken cost PHP 640. How many kilos of chicken can you buy with PHP 3 200?
Solution: The ratio of the kilos of chicken that can be bought to the cost is 4 : 640. Now, let’s use N to represent the number of kilos of chicken that can be bought with PHP 3200. Thus, we have the ratio N : 3200.
4 : 640 = N : 3200
Let us apply the fact that the product of the means is equal to the product of extremes so we can determine N.
Multiplying the means of the ratio: 640 x N
Multiplying the extremes of the ratio: 4 x 3200 = 12800
Since the product of the means is equal to the product of the extremes:
640 x N = 12800
What must be multiplied to 640 to obtain 12800? We determine that number by dividing 12800 by 640.
N = 12800 ÷ 640 = 20
Therefore, you can buy 20 kilos of chicken with PHP 3200.
Example: If 5 : 4 = 35 : 28, what should be N so that 4 : 5 = 28 : N
Solution: Since the ratios in the proportion are reciprocated, we can use the second property of proportions. Using the second property, N = 35.
Example: When A is divided by 5, the result will be equal to the result when you divide B by 2. What is the result if you divide A by B?
Solution: The problem sounds tricky since we have no idea what the values of A and B are. However, using the third property of proportion, we can determine the result when we divide A by B.
A divided by 5 can be written as A⁄5, which can then be expressed into a ratio as A : 5.
Meanwhile, B divided by 2 can be written as B⁄2, which can then be expressed into a ratio as B : 2.
Since the problem states that if A is divided by 5, the result will be equal to the result if B is divided by 2, then
A : 5 = B : 2
We want to know, what will be the result when we divide A by B or A⁄B or, as a ratio, A : B
So, from A : 5 = B : 2, how can we obtain A : B?
We can apply the property that if we switch the means of a proportion, the result is still a proportion.
Let us now switch the means of A : 5 = B : 2
ratio and proportion 8
We obtain A : B = 5 : 2. Expressing into a fractional form:
A⁄B = 5⁄2
Therefore, if A is divided by B, the result is 5⁄2 or 2.5.
