Understanding Ratios
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About Understanding Ratios
In these notes you are going to be able to Understand Ratios when calculating and solving questions. Make sure to practice all the questions so that you get used to solving problems of this nature.
Note that “Understanding Ratios” is a sub topic and you can also access more notes, questions and answers under the main topic “Operations On Numbers And Calculator Skills“ which is one of the recommended learning topics in the Mathematical Literacy Curriculum.
What is a ratio?
A ratio is a way to compare two quantities by showing the relative size of one quantity to another. It is expressed as a quotient of two numbers, which can be written in several different forms:
\[1.\;As\;a\;fraction:\frac ab\;\;\\2.\;U\sin g\;a\;colon:\;a:b\;\\3.\;In\;words:\;”a\;to\;b”\]Example:
If there are 10 boys and 15 girls in a class, the ratio of boys to girls is
, which simplifies to
What is are equivalent ratios
Equivalent ratios are ratios that express the same relationship between quantities, even though they may use different numbers. Two ratios are equivalent if they have the same simplest form or if their cross products are equal.
Ratios:
and
Simplification:
\[\frac23and\frac46\\Simplify\;\frac23and\frac46Since\;both\;simplify\;to\frac23\\they\;are\;equivalent.\]Cross Multiplication:
3×4=12
Show calculations for:
- 7 : 6 = 35 : 30
- 33 : 12 = 11 : 4
Answer:
1.
\[7\;:\;6\;=\;35\;:\;30\\=\frac76\\=\frac{7\times5}{6\times5}\\=\frac{35}{30}\]2.
\[33\;:\;12\;=\;11\;:\;4\\=\frac{33}{12}\\=\frac{33\div3}{12\div3}\\=\frac{11}4\]Simplifying ratios
example questions
- Given that John earns R12,000 per month and Sam earns R15,000 per month, what is the ratio of their incomes?
- Write the ratio 15,4: 96,6 in a simplified form.
- Express the following in its simplest form:
\[\frac56\;:\;\frac34\] - 15 minutes : 1 hour
- 1,2 m : 35 cm
Answer:
1.
\[John\;:\;Sam\\12\;000\;:\;15\;000\\12\;:\;15\\4\;:\;5\\\]2.
\[15,4\;:\;96,6\\154\;:\;966\\11\;:\;69\]3.
\[\frac56\;:\;\frac34\\\frac{20}{24}\;:\;\frac{18}{24}\\20\;:\;18\\10\;:\;9\]4.
15 minutes : 60 minutes
15 : 60
1 : 4
5.
1,2 m : 35 cm
120 cm : 35 cm
120 : 35
24 : 7
Missing Value Ratio Problems
When dealing with this kind of question, you are provided with the ratio of two quantities and the value of one of them. Your task is to determine the unknown value of the other quantity.
Example:
Method 1 (Multiplication/Division):
Given a candy manufacturer who needs to fill a box with green and yellow sweets in a 3:7 ratio, if there are 12 green sweets in the box, how many yellow sweets should be added to maintain the specified ratio?
-
Determine the total parts in the ratio:
The total parts are . -
Calculate the value of one part:
Since there are 12 green sweets, which correspond to 3 parts in the ratio, we can find the number of sweets per part by dividing the total number of green sweets by the number of parts representing green sweets.
\[Number\;of\;sweets\;per\;part=\frac{12}3=4\\\] - Calculate the number of yellow sweets:
The yellow sweets correspond to 7 parts. Therefore, the number of yellow sweets is:
Number of yellow sweets = 7 × 4 = 28
Method 2 (Determining the value of 1 part)
Given that Ronald and Ben are investing in a Fast Food business in a ratio of 4:7, and Jim’s investment is R182,000, how much does Ben invest in the business?
1. Ben and Jim’s Investment Ratio
Ben and Jim have an investment ratio of 4:7. Jim’s investment amounts to R182,000, which corresponds to 7 parts of the total ratio.
- Calculate the value of one part.
- Determine how much Ben invests if he puts in 4 parts.
2. Solution Steps:
- Find the value of one part by dividing Jim’s investment by 7.
- Multiply the value of one part by 4 to find Ben’s total investment.
3. Work:
- Jim’s 7 parts = R182,000
- Therefore, one part = R182,000 ÷ 7 = R26,000
- Ben invests 4 parts: 4 × R26,000 = R104,000
Sharing and dividing in a given ratio
In this type of problem, we are given the ratio of two quantities and the sum total of both quantities. We therefore need to share the sum total into the given ratios. There are two methods to solve this type of problem:
John, Sam and Smith make R1 200 profit from selling cakes.
Determine how much profit each girl receives, if they invested in
the ratio of 1 : 2 : 3.
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Determining the value of 1 part
1. A Grade 12 class has 12 boys and 36 girls. Give the ratio of boys to girls in this class.
boys: girls
12 : 36
HCF = 12
1 : 3
2. Express the ratio 12,4 : 4,8 in its simplest form.
12,4 : 4, 8
124 : 48
31 : 12
3. Simplify
\[\frac2{21}\;:\;\frac5{63}\\\frac2{21}\;:\;\frac5{63}\\\frac6{63}\;:\;\frac5{63}\\6\;:\;5\\\\\\\]4. How can you express the scale of a map where 1 centimeter represents 100 kilometers in reality? What is the corresponding map-to-real-life ratio?
map : real-life
1 cm : 100 km
1 cm : 10 000 000 cm
1 : 10 000 000
5. Given that cement, stone, and sand are to be mixed in a ratio of 4 : 3 : 1 for a specific construction project, and it is known that 900 grams of stone is needed, determine the required amounts of cement and sand.
Multiplication/Division (Method 1)
cement : stone : sand
4 : 3 : 1
? : 900g : ?
Mass of cement = 4 x 300 g = 1 200 g
Mass of sand = 1 x 300 g = 300 g
Answers Tip: 3 x the number we want to find = 900 g
3 X 300 g = 900 g
Determining the value (Method 2)
cement : stone : sand
4 : 3 : 1
? : 900 g : ?
∴The mass of 900 g of stone represents 3 parts
∴1 part = 900 g + 3 = 300 g
∴Mass of cement = 4 parts x 300 g = 1 200 g
∴Mass of sand = 1 part x 300 g = 300 g
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Mixed Exercises
1. Simplify the following ratios:
1.1 12: 960
1.2 140: 24
1.3 12,3: 6,6
1.4 12 seconds : 1 hour
1.5 15 cm: 0,65 m
1.6 \[\frac38\;:\;\frac25\]
2. Given that Joseph’s monthly income exceeds Mandy’s by R15,000, with Mandy earning R10,000 per month, what is the ratio of Joseph’s income to Mandy’s income?
3. Mandy and Joseph’s heights are in the ratio 5: 7. If Joseph is 189 cm tall, how tall is Mandy?
4. Given that one shareholder invested R300,000 and another invested R500,000 in a company, and the company earned an annual profit of R1,200,000, calculate the share of the profit that each shareholder should receive in proportion to their respective investments.
5. Given: The optimal proportion of students to teachers is 25 to 1.
-
- What is the student-to-teacher ratio in a school with 860 students and only 20 teachers?
- Given that School A has 750 students and 15 teachers, and School B has 920 students with 20 teachers, which institution offers a smaller number of students per teacher?
Answer:
1.1
12 : 960
1 : 80
÷ HCF = 12
1.2
140 : 24
35 : 6
÷ HCF = 4
1.3
12,3 : 6,6
x10 to get rid of decimal
123 : 66
÷ HCF = 3
41 : 22
1.4
12 s: 1 h
12 s: 3 600 s
12 : 3 600
∴devide it by 12
1 : 300
Hints
. . . 1 h x 60 = 60 min
60 minx 60
= 3 600 seconds
1.5
15 cm : 0,65 m
15 cm: 65 cm
15: 65
Divide by 5
3: 13
Hint
. . . 0,65 m x 100 = 65 cm
1.6
\[\frac{15}{40}\;:\;\frac{16}{40}\\15\;:\;16\\\\Hints:\\\frac{3\times5}{8\times5}=\frac{15}{40}and\frac{2\times8}{5\times8}=\frac{16}{40}\\Common\;Denominator\;is\;40\\\\\]
2.
Joseph's : Mandy's
R25 000 : R10 000
25 000 : 10 000
Hint+ HCF = 5 000
5:2
3.
Sally : David
5:7
Hint: x 27
𝑥 : 189 cm
Mandy's height
= 5x27
= 135 cm
OR
Sally : David
5 : 7
𝑥 :189 cm
𝑥= 189 cm + 7
= 27
Mandy's Height
= 5 parts x 27
= 135 cm
4.
\[R300\;000\;:\;R500\;000\\3\;:\;5\\Total\;number\;of\;parts\;=\;3\;+\;5\;=\;8\;parts\\Shareholder\;A:\frac38\times\frac{1\;200\;000}1=\;R450\;000\\Shareholder\;B:\frac58\times\frac{1\;200\;000}1=\;R750\;000\]
OR
R300 000 : R500 000
3:5
Total number of parts = 3 + 5 = 8 parts
1 Part = R1 200 000 + 8 = R150 000
Shareholder A: R 150 000 x 3 parts = R450 000
Shareholder B: R 150 000 x 5 parts = R750 000
5.1
learners : teachers
Hint:multiply by 32
800:?
Number of teachers
= 1 X 32
= 32 teachers
OR
learners : teachers
25: 1
800:?
1 Part = 800 + 25
= 32
Number of teachers
= 1 X 32
= 32 teachers
5.2
learners : teachers
860 : 20
Hint: ÷ HCF = 20
5.3
School A
learners : teachers
750 : 15
50: 1
School B
learners : teachers
920: 20
46: 1
School B has a lower number of students per teacher.
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