Grade 9 Ratio and Proportion

Definition: A ratio is a comparison of two quantities of the same unit by division. It shows how many times one quantity is of the other.

Notation: The ratio of two quantities 'a' and 'b' can be written as a:b, or as a fraction a/b. The first term 'a' is called the antecedent, and the second term 'b' is called the consequent. The consequent 'b' cannot be zero.
Example: The ratio of 3 apples to 5 oranges is 3:5 or 3/5. Here, 3 is the antecedent and 5 is the consequent.

Simplification: Ratios are usually expressed in their simplest form by dividing both the antecedent and the consequent by their greatest common divisor (GCD).
Example: The ratio 6:8 can be simplified by dividing both by their GCD, which is 2. So, 6:8 is equivalent to 3:4.

Units: The quantities being compared in a ratio must have the same units. If they have different units, convert them to the same unit before forming the ratio.
Example: Ratio of 50 cm to 2 meters. Convert 2 meters to 200 cm. The ratio is 50 cm : 200 cm, which simplifies to 1:4.

Definition: A proportion is an equality of two ratios. If the ratio a:b is equal to the ratio c:d, then a, b, c, and d are in proportion. This is written as a:b :: c:d or a/b = c/d.

Terms of a Proportion: In the proportion a:b :: c:d:

• 'a' and 'd' are the **extremes** (outer terms).
• 'b' and 'c' are the **means** (middle terms).

Product of Extremes and Means: A fundamental property of proportion is that the product of the extremes is equal to the product of the means.
If a/b = c/d, then a * d = b * c. This is also called cross-multiplication.

Checking Proportion: To check if two ratios form a proportion, simplify both ratios to their simplest form and see if they are equal, or use the product of extremes and means property.
Example: Are 2:3 and 6:9 in proportion?
Method 1 (Simplification): 2:3 is already simple. 6:9 simplifies to 2:3 (dividing by 3). Since the simplified ratios are equal, they are in proportion.
Method 2 (Cross-multiplication): 2/3 = 6/9. Check if 2 * 9 = 3 * 6. 18 = 18. Since the products are equal, they are in proportion.

Step 1: Set up the proportion with the missing term represented by a variable (e.g., x).
Problem: Find the missing term in the proportion 4:5 :: x:20.
Write as fractions: 4/5 = x/20.

Step 2: Use the property that the product of the extremes equals the product of the means (cross-multiplication).
4 * 20 = 5 * x

Step 3: Simplify and solve the resulting equation for the variable.
80 = 5x

Step 4: Isolate x by dividing both sides by 5.
x = 80 / 5

Step 5: Calculate the value of x.
x = 16

Result: The missing term is 16. The proportion is 4:5 :: 16:20.

If a/b = c/d, then:

Invertendo: If a/b = c/d, then b/a = d/c (provided a and c are not zero). You can invert both ratios.
Example: If 2/3 = 4/6, then 3/2 = 6/4 (which is also true).

Alternendo: If a/b = c/d, then a/c = b/d (provided b and c are not zero). You can alternate the means.
Example: If 2/3 = 4/6, then 2/4 = 3/6 (which simplifies to 1/2 = 1/2, also true).

Componendo: If a/b = c/d, then (a+b)/b = (c+d)/d.
Example: If 2/3 = 4/6, then (2+3)/3 = (4+6)/6 => 5/3 = 10/6 (which simplifies to 5/3 = 5/3, true).

Dividendo: If a/b = c/d, then (a-b)/b = (c-d)/d (provided b and d are not zero).
Example: If 2/3 = 4/6, then (2-3)/3 = (4-6)/6 => -1/3 = -2/6 (which simplifies to -1/3 = -1/3, true).

Componendo-Dividendo: If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d) (provided a is not equal to b, and c is not equal to d).