We use ratios to make comparisons between two things. When we express ratios in words, we use the word "to" – we say "the ratio of something to something else" Ratios can be written in several different ways.

Multiplying or dividing each term by the same nonzero number will give an equal ratio. For example, the ratio 2:4 is equal to the ratio 1:2. To tell if two ratios are equal, use a calculator and divide. If the division gives the same answer for both ratios, then they are equal.

How can we write the ratio of squares to circles, or 3 to 6? The most common way to write a ratio is as a fraction, 3/6. We could also write it using the word "to," as "3 to 6.“ Finally, we could write this ratio using a colon between the two numbers, 3:6. Be sure you understand that these are all ways to write the same number.

ratio of squares to circles is 3/6 ratio of squares to circles is 3 to 6 ratio of squares to circles is 3:6

Some other equal ratios: 3:6 = 12:24 = 6:12 = 15:30 Are 3:12 and 36:72 equal ratios? Find 3÷12 = 0.25 and 36÷72 = 0.5 The quotients are not equal —> the ratios are not equal

You can also use decimals and percents to compare two quantities. In our example of squares to circles, we could say that the number of squares is "five- tenths" of the number of circles, or 50%.

EXAMPLE 1: REDUCE 36 : 72 TO THE SIMPLEST FORM. SOLUTION: GIVEN: RATIO 36 : 72. TO FIND: SIMPLEST FORM OF RATIO. HCF OF 36 AND 72 IS : 72 = 36/72 (DIVIDE EACH TERM BY HCF I.E. 36) = (36/36)/(72/36) (SIMPLIFY) = 1/2 = 1 : 2 ANSWER: RATIO 36 : 72 IN SIMPLEST FORM IS 1 : 2.

Example 2: Find the simplest form of ratio of 1.5kg to 500g. Solution: Given: 1.5kg to 600g. To find: Simplest form of ratio. Ratio of 1.5kg to 600g = 1.5kg/600g (convert kg to grams).** = (1.5 x 1000)g/600g. = 1500/600 (divide both terms by HCF i.e. 300). = 5/2. = 5 : 2. Answer: Required ratio is 5 : 2.

Example 3: Which ratio is greater 2: 3 or 5 : 7. Solution: Given ratios 2 : 3 and 5 : 7. To find: Which ratio is greater. First ratio 2 : 3 = 2/3. Second ratio 5 : 7 = 5/7. LCM of denominators 3 and 7 is 21. Multiply both ratios by 21. Therefore, First ratio = 2/3 x 21 (simplify). = 2 x 7. = 14. Second ratio = 5/7 x 21 (simplify). = 5 x 3. = 15. Since 15 > 14, Ratio 5 : 7 > 2 : 3

Example 4: Are the ratios 3 to 4 and 6 : 8 equal? Solution: Given: Ratios 3 : 4 and 6 : 8 To find: Are they equal. Method I: First ratio 3 : 4 = ¾. Second ratio 6 : 8 = 6/8 (divide both terms by HCF i.e. 2). = 3/4 Each ratio is 3/4. Therefore, the ratios are equal. Answer: The ratios are equal. Method II: The ratios are equal if 3/4 = 6/8. These are equal if their cross products are equal. Therefore, 3 × 8 = 4 × = 24. Since both of these products are equal to 24, the ratios are equal.

Example 5: Simplify the ratio: 2/3 : 4/7 : 3. Solution: Given: 2/3 : 4/7 : 3. To find simplest form Ratio 2/3 : 4/7 : 3 = 2/3 : 4/7 : 3/1**. LCM of denominators 3, 7, and 1 is 21. Multiply each term of the given ratio by 21. Therefore, Ratio 2/3 : 4/7 : 3/1 = (2/3 x 21) : (4/7 x 21) : (3/1 x 21). = (2 x 7) : (4 x 3) : (3 x 21) = 14 : 12 : 63 Answer: Simplified ratio is 14 : 12 : 63. **While solving fractions you should write whole numbers in fraction form dividing it by 1 e. g. 3 (whole number) = 3/1 (fraction).

Example 6: Joliet has a bag with 8 mangoes, 14 apples, 12 oranges and 1 pine-apple. What is the ratio of mangoes to oranges? Solution: Step 1: Write what is given and what is to be found. Given: Mangoes 8, apples 14, oranges 12 and pineapple 1. To find: Ratio between mangoes and oranges. Step 2: Write the expression. (Hint: Write the numerator equal to the first quantity and the denominator equal to the second quantity. In this case first quantity is mangoes). Ratio of mangoes to oranges = Number of mangoes/Number of oranges.

Step 3: Substitute the values of mangoes and oranges. Ratio of mangoes to oranges = 8/12. Step 4: Simplify the expression. (Hint: Divide both numerator and denominator by HCF to obtain simplest form). Ratio of mangoes to oranges = 8/12 (divide both terms by HCF i.e. 4).** = 2/3. = 2 : 3. Step 5: Write the answer. Ratio of mangoes to oranges is 2 : 3. (** HCF of 8 and 12 is 4) Note: The answer can be written as: Ratio is 2 : 3 or 2/3 or 2 to 3