Chapter Notes: Ratio, Proportion and Unitary Method
Introduction
Ever wondered how we compare things in our daily lives, like the speed of a car versus a bike, or how to fairly divide a pizza among friends? The chapter on Ratio, Proportion, and Unitary Method is like a mathematical toolkit that helps us make sense of comparisons and calculations in a simple and organized way. Whether it's splitting pocket money, figuring out how long a task takes, or understanding how quantities relate to each other, this chapter introduces concepts that are not only useful in math but also in real-life situations. Let's dive into the fascinating world of ratios, proportions, and the unitary method to see how they make our lives easier!
Ratio
- Ratio is a way to compare two quantities of the same kind by showing how many times one quantity is of the other.
- It is written as a:b, read as "a is to b," where 'a' is the antecedent (first term) and 'b' is the consequent (second term).
- Both quantities must be in the same units for comparison.
- The ratio a:b is equivalent to the fraction a/b.
Example: Reema scored 70 marks, and Meenu scored 50 marks in an English test. The ratio of Reema's marks to Meenu's marks is 70:50. Simplify by dividing both by their greatest common divisor (10): 70 ÷ 10 = 7 and 50 ÷ 10 = 5. Thus, the ratio is 7:5.
Simplest Form of Ratio
- A ratio is in its simplest form when its terms are co-prime, meaning they have no common factors other than 1.
- To simplify, divide both terms by their greatest common divisor (GCD).
Example: Simplify the ratio 20:15. Find the GCD of 20 and 15, which is 5. Divide both terms by 5: 20 ÷ 5 = 4 and 15 ÷ 5 = 3. Thus, the simplest form is 4:3.
Equivalent Ratios
- Equivalent ratios are ratios that represent the same relationship and are obtained by multiplying or dividing both terms of a ratio by the same non-zero number.
Example: The ratio 1:2 is equivalent to 2:4 (multiply both terms by 2) or 5:10 (multiply both terms by 5).
Points to Remember
- Ratios compare quantities of the same kind only.
- Quantities must be in the same unit before finding the ratio.
- The order of terms matters: a:b is different from b:a.
- Multiplying or dividing both terms by the same non-zero number does not change the ratio's value.
Example: Find the ratio between 35 kg and 45 kg. The ratio is 35:45. Simplify by dividing by the GCD (5): 35 ÷ 5 = 7 and 45 ÷ 5 = 9. Thus, the ratio is 7:9.
Comparison of Ratios
- To compare ratios, ensure they have the same consequent or convert them to fractions with a common denominator.
- Compare the numerators of the fractions to determine which ratio is larger.
Example: Compare the ratios 3:8, 1/2:4, and 5/6:2/3. Simplify 1/2:4 to 1:8 (multiply both by 2). Simplify 5/6:2/3 to 5:4 (multiply by 6). Convert all to have consequent 8: 3:8 = 3:8, 1:8 = 1:8, 5:4 = 10:8 (multiply by 2). Compare 3, 1, 10; since 10 > 3 > 1, the ratio 5/6:2/3 is the greatest.
Compound Ratio
- A compound ratio combines two or more ratios by multiplying their antecedents for the new antecedent and their consequents for the new consequent.
- For ratios a:b and c:d, the compound ratio is ac:bd.
Example: Find the compound ratio of 7:3 and 1/5:3/10. Simplify 1/5:3/10 to 2:3 (multiply by 10). Compound ratio = (7 × 2):(3 × 3) = 14:9.
Continued Ratio
- Three quantities a, b, c are in continued ratio if a:b = b:c, written as a:b:c.
- Used to compare three quantities of the same kind.
Example: If p:q = 21:25 and q:r = 25:27, then p, q, r are in continued ratio, written as 21:25:27.
Increase or Decrease in a Given Ratio
- When a quantity changes, the ratio of the old quantity to the new quantity is used to find the multiplying ratio.
- Final quantity = (b/a) × Original quantity, where a:b is the ratio of old to new quantity.
- Multiplying ratio = Final quantity / Original quantity.
Example: Find the multiplying ratio that changes 96 kg to 69 kg. Multiplying ratio = 69/96 = 23/32.
To Divide a Given Quantity in a Given Ratio
- Dividing into two parts: For a quantity A divided in the ratio x:y, first part = (x/(x+y)) × A, second part = (y/(x+y)) × A.
- Dividing into three parts: For a quantity A divided in the ratio x:y:z, first part = (x/(x+y+z)) × A, second part = (y/(x+y+z)) × A, third part = (z/(x+y+z)) × A.
Example: Divide 230 kg mangoes among X, Y, Z in the ratio 1/2:1/3:1/8.
- Simplify the ratio 1/2:1/3:1/8 to 12:8:3 (multiply by LCM 24).
- Shares: X = (12/23) × 230 = 120 kg,
- Y = (8/23) × 230 = 80 kg,
- Z = (3/23) × 230 = 30 kg.
Proportion
- Four quantities p, q, r, s are in proportion if p:q = r:s, written as p:q::r:s, read as "p is to q as r is to s."
- p and s are extremes, q and r are means; the product of extremes equals the product of means (ps = qr).
- The first two and last two quantities must be of the same kind, but the pairs can differ.
Example: Check if 18, 14, 45, 35 are in proportion. 18:14 = 9/7, 45:35 = 9/7. Since 18:14 = 45:35, they are in proportion.
Continued Proportion
- Three quantities p, q, r are in continued proportion if p:q = q:r, i.e., q² = pr.
- q is the mean proportional, r is the third proportional to p and q.
Example: Find the mean proportional between 72 and 8. Let x be the mean proportional. Then 72:x = x:8, so x² = 72 × 8 = 576, x = 24.
Unitary Method
- The unitary method finds the value of one unit of a quantity to calculate the value of a required quantity.
- Steps: Divide to find the value of one unit, then multiply to find the value of the required quantity.
Example: If 5 books cost ₹335, find the cost of 7 books. Cost of 1 book = 335/5 = ₹67. Cost of 7 books = 67 × 7 = ₹469.
Variation
- Variation describes how one quantity changes with respect to another.
- Two types: direct variation and inverse variation.
Direct Variation
- In direct variation, an increase (or decrease) in one quantity causes a corresponding increase (or decrease) in another.
- Use division to find the unit value, then multiply for the required value.
Example: A man earns ₹15,750 in 15 days. Find earnings in 28 days. Earnings per day = 15,750/15 = ₹1,050. Earnings in 28 days = 1,050 × 28 = ₹29,400.
Inverse Variation
- In inverse variation, an increase in one quantity causes a decrease in another, and vice versa.
- Use multiplication to find the unit value, then divide for the required value.
Example: Food lasts 63 days for 26 students. For how many students will it last 39 days? For 1 day, food lasts for 26 × 63 students. For 39 days, it lasts for (26 × 63)/39 = 42 students.
Time and Work
- Time and work problems use the concept: One day's work = 1 / (Number of days to complete the work).
- Number of days = 1 / (One day's work).
- Time to complete work = (Work to be done) / (Work done in unit time).
Example: A and B complete a work in 20 and 30 days, respectively. Find time to complete together.
- A's one day's work = 1/20, B's = 1/30.
- Together = 1/20 + 1/30 = 5/6.
- Time together = 1 / (5/6) = 6/5 = 1 1/5 days.
FAQs on Chapter Notes: Ratio, Proportion and Unitary Method
| 1. What is the difference between ratio and proportion? |  |
| 2. How do you solve problems using the unitary method? | |
| 3. What are the applications of ratios and proportions in real life? | |
| 4. Can you explain the concept of variation and its types? | |
| 5. How does the concept of time and work relate to ratios and proportions? | |
