Mathematics (New)
RATIOS AND PROPORTIONS
RATIOS AND PROPORTIONS
This topic covers two concepts (Ratios and Proportions) in a detailed manner.
Upon completion of this chapter, you should demonstrate competencies in Ratios and Proportions by (i) Expressing Ratios in their simplest forms and (ii) Dividing a given quantity into Proportional parts correctly.
These competencies should enable you to use ratios and proportions in daily life; such as preparing balanced diets as prescribed by dieticians, sharing and distributing of useful resources for example money and land in right shares or right proportions.
Introduction
In our daily lives, we often compare quantities. For example, we compare the number of boys to that of girls in a class, the amount of sugar to flour in a recipe and the distance traveled to the time taken. Ratios and proportions help us make these comparisons clearly and accurately.
Ratios
A ratio shows the relationship between two or more quantities by indicating how many times one quantity contains another. Ratios can compare quantities of the same kind or different kinds, provided they are expressed in the same units.
The Concept of Ratios and Solve Related Problems.
Explain the Concept of Ratios and Solve Related Problems.
A ratio is a way of comparing quantities measured in the same units and expressed in a specific order. It can be written in the form of 𝑋:𝑌 or as a fraction 𝑋/𝑌 and normally expressed in a simplest form.
Examples of ratios include the following;
(a) A class with 45 girls and 40 boys. The ratio of number of boys to the number of girls is 40:45
(b) A football ground 100𝑚 long and 50𝑚 wide. The ratio of length to the width is 100:50
Note that ratios can be written in its simplest form like fractions;
For example, the ratio 40:45 = 40/45 which simplifies into 8/9 or 8∶9. Similarly 100:50 = 100/50 = 2∶1
Example 41
Express the following comparisons as ratios;
(a) An oak tree 20m high to a pine tree 60m high.
Solution;
(a) The height ratio of an oak tree to a pine tree is 20:60
(c) The weight ratio of a dog to that of a boy is 21:35
Example 42
Simplify the following ratios, giving your answers as whole numbers:
Solution;
Example 43
Express the following ratios in the form of 𝑘∶1
(a) 0.8 ∶1.6 (b) 55 ∶11
Example 44
A special cereal mixture contains rice, wheat, and corn in the ratio 2:3:5, respectively. If a bag of the mixture has 3Kg of rice, what amount of corn does it contain?
Solution;
Exercise 13
1. Express the following ratios in their lowest terms;
(a) 8:12 (b) 40:50 (c) 14∶ 0.5
(d) 7:0.007 (e) 3.5:0.07
2. Find 𝑥 in each of the following;
(a) 1:2 = 6∶𝑥 (b) 5:𝑥 = 7∶21
(c) 8:𝑥 = 20:15 (d) 3𝑥:8 =7:24
3. Two squares have sides 12cm and 15cm in length. Find the ratios of:
(a) Their perimeters.
(b) Their areas.
4. In a mixed school there are 110 students. If 44 of these are girls, find;
(a) The ratio of boys to girls in the school.
6. A drove of 63 pigs consists of white and black pigs. If it contains 23 white pigs, find the ratio of black pigs to white pigs.
Proportions
Proportions express equal ratios and show how quantities are connected and change together in a consistent way.
The Concept of Proportions
Explain the Concept of Proportions
The concept of Proportions is important in solving daily life problems especially those involving scaling, ratios and percentages.
Such problems compare different quantities. Therefore, a proportion is a part or a number considered in comparison to a whole.
However, in the context of ratios, a proportion stands as a statement showing that two ratios are equal. For example, if 2 mathematics books cost Tsh. 10,000, then the same 4 mathematics books cost Tsh. 20,000.
Proportions can be written as ratios or fractions, that is 𝑎:𝑏 = 𝑐:𝑑 is the same as 𝑎/𝑏 = 𝑐/𝑑, from which we can make a cross multiplication and get 𝑎×𝑑 = 𝑏×𝑐.
Proportional Parts and Solve Related Problems
Explain Proportional Parts and Solve Related Problems
A proportional part is a portion of a total quantity that is allocated according to a specified ratio or proportion. It is a share of a total amount that matches a given ratio.
For example, if we want to divide a quantity into two parts which are in the ratio say 2:3, we make 5 equal divisions (2+3=5). The required parts will then be 2 and 3 times these divisions, that is 2/5 𝑎𝑛𝑑 3/5.
Example 45
Divide £ 77 between two people in the ratio 6:5;
Solution;
When dividing in the ratio 6:5 we make 11 parts.
Thus, the first person gets 6/11 of £77 = (6/11)×(77/1)=£42, the second person gets (5/11)×(77/1)= £35
∴ When two people share £77 in the ratio 6:5, the first person gets £42 and the second person gets £35.
Note that a quantity can be divided or shared in more than two parts under specified ratios.
Example 46
Find the proportional parts of 156 in the ratios 3:4:5
Solution;
Total number of parts =156, the sum of terms of the ratios is 3+4+5 = 12
The required proportional parts are (3/12)×156 = 39, (4/12)×156 = 52 and (5/12)×156 = 65,
∴ The required proportional parts are 39, 52 and 65.
Example 47
Eradia, Faith and Richard were given an amount of 900,000/− to share in the ratio of 2∶3∶4 . How much did each one get?
Solution;
Total ratio = 2+3+4 = 9,
The amount for Eradia is (2/9 )x 900,000 = 200,000/−
The amount for Faith (3/9) x 900,000 = 300,000/−
The amount for Richard is (4/9) x 900,000=400,000/−
∴ Eradia, Faith and Richard got 200,000/− , 300,000/− and 400,000/− respectively.
Example 48
John and Ashura shared 40,000 Tanzanian shillings in the ratio 3:5, how much money did each get?
Solution;
The sharing ratio is 3:5 = 3/5, where 3 is for John and 5 is for Ashura;
The sum of terms of the ratios is 3+5 = 8,
So, the share for John is (3/8)×40000 =15,000 and that for Ashura is (5/8)×40000 =25,000
Therefore, John got 15,000 Tsh while Ashura got 25,000 Tsh.
Example 49
The angles of the triangle are in the ratio 4∶5∶9. Find the largest angle of the triangle.
Solution;
Total ratio = 4+5+9 =18
Since any triangle has a sum of angles 180°,
Then the largest angle is at a large ratio value which is 9, and so the angle is (9/18) x 180° = 90°
∴ The largest angle of the triangle is 90°.
Example 50
The first, second, third and fourth terms are proportional. If the first, second and third terms are 42, 36 and 35 respectively, find the fourth term.
Solution;
Let 𝑥 be the third term and so the proportional terms become 42, 36, 35 and 𝑥.
Their corresponding ratios are (42/36) = 35/𝑥, or 42𝑥 = 36×35, solving for 𝑥 gives 𝑥 = (36×35)/42 = 30
Therefore, the fourth term is 30.
Exercise 14
1. Divide Tsh. 110,000 between two people in the ratio 6:5.
2. Patric, Amina and Johari shared some mangoes. The ratio of number of mangoes received by Patrick and Amina was 7:3. The ratio of number of mangoes received by Patrick and Johari was 4:5. If Johari received 21 more mangoes than David, what is the number of mangoes received by Amina?
3. A powdery mixture is made up of powders A and B in the ratio 5:7, if the mixture made has120 Kg, how much of each type of powder is contained in it?
4. Abdalah, Asha and Said were given Tsh 240,000 Tanzanian shillings by their father so that they share it in the ratio 2:4:6, how much money did Asha get?
5. The angles of the triangle are in the ratio 5∶6∶7. Find the smallest angle of the triangle.
6. Mr. Kaloviyo gave his piece of land of 23 hectares to his four children (Baraka, Beatrice, Donatus and Magdalena) so that they share it in the ratio 1:2.5:3:4, what size of land did each get?
Topic Summary
Important Hints about Ratios and Proportions
1. A ratio is a comparison between two or more quantities having the same units.
2.The ratio of 𝑥 to 𝑦 is written as 𝑥:𝑦 which is the same as 𝑥/𝑦.
3. A proportion is a part or a number considered in comparison to a whole.
4. In relation to ratios, a proportion stands as a statement showing that two ratios are equal.
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