5.1 : Ratio of Two Quantities Use ratios To determine if given ratios are equivalent Write ratios in their simplest form Ratio is a comparison between two quantities of the same kind Ratio of two quantities, a and b, is written as a:b a b Read as ‘a is to b’ What is the ratio of women to men? 3:2 What is the ratio of men to women? 2:3 What is the ratio of blue balls to pink? 9:3 or 3 : 1 What is the ratio of pink balls to blue? 9:6 or 3 : 2 An equivalent ratio can be obtained when we multiply or divide both quantities in a ratio by the same number 2 1 x3 x3 6 = 3 8 ÷4 2 ÷4 = 12 3 2:3 6:9 A proportion is an equation that states that two ratios are equal, such as: In simple proportions, all you need to do is examine the fractions. If the fractions both reduce to the same value, the proportion is true. Are these proportional? 5 2 and 15 6 55 1 15 5 3 22 1 62 3 This is a true proportion, since both fractions reduce to 1/3. In simple proportions, you can use this same approach when solving for a missing part of a proportion. Remember that both fractions must reduce to the same value. To determine the unknown value, you must cross multiply. (3)(x) = (2)(9) 3x = 18 x=6 Check your proportion (3)(x) = (2)(9) (3)(6) = (2)(9) 18 = 18 True! 2 10 3 5 4 x 6 12 No x=8 25 5 x 2 x=10 a) If 4 tickets to a b) A house which is appraised for $10,000 show cost $9.00, pays $300 in taxes. find the cost of 14 What should the tax be tickets. on a house appraised at $15,000. =$31.50 =$450 A house painter mixes yellow paint with blue paint to get green paint. The ratio of the volume of the yellow paint to the volume of the blue paint is 5:3. if the difference in volumes of the two paints is 500ml, find the volume of the green paint. 5 : 3 + = 4 500 5 3 ml Difference in volumes between yellow and blue 5 3ml 2000 = 500ml Volume of green paint Difference in volumes between yellow and blue Volume of green paint 500 ml Volume of green paint 53 = 53 8 = 2 4 500ml 2000ml Ratio of three quantities is a comparison of three quantities having the same unit of measurement Eg: The ages of three children are 10 years, 11 years and 13 years. Ratio of their ages is 10:11:13 REMEMBER? Unit is not shown in the ratio Are these equivalent? 1:5:7 and 4:20:28 yes 1 : 5 : 7 x4 : x4 : x4 4 : 20 : 28 = 4 : 20 : 28 Are these equivalent? No 2:3:5 and 10:14:20 2 : 3 : 5 x5 : x5 : x5 10 : 14 : 20 ≠ 10 : 15 : 25 Whole Number Divide each by HCF Fractions or mixed number Multiply each by LCM REMEMBER! Ratio in lowest terms must be in whole number Decimals Multiply each term by 10, 100, 1000 .. Whole Number Divide each by HCF 4 : 20 : 28 ÷4 : ÷4 : ÷4 1 : 5 : 7 Simplify 6 : 72 : 42 Fractions or mixed number Multiply each by LCM 1 1 : :3 2 4 x4 : 1 4 2 2 x4 : x4 Simplify 2 1 1 : : 3 2 6 1 4 3 4 4 : 1 : 12 Answer! Multiply each term by 10, 100, 1000 .. Decimals 0.1 : 2 : 0.036 x1000 : x1000 : x1000 100 : 2000 : 36 Simplify 0.3: 0.0045: 0.24 Snakes : cats 2 : 3 Cats : Tigers 3 : 5 Snakes : Cats : Tigers 2 : 3 : 5 Adam’s pet Red : Blue 1 : 2 x2 : x2 Blue : Yellow 4 : 7 Red : Blue : Yellow 2 : 4 : 7 Faiz’s marbles Given the ratio of a bungalow to a semi-detached house to a terrace house is 5 : 3 : 2. If the price of bungalow is RM 1 200 000, what is the price for a semi-detached house? Unitary method Proportional method Bungalow : Semi-D : Terrace 5 : 3 : 2 5 parts represent RM 1 200 000 1 part represents = RM 1 200 000 5 = RM 240 000 Semi-D 3 parts represent = RM 240 000 x 3 = RM 720 000 Terrace 2 parts represent = RM 240 000 x 2 = RM480 000 Bungalow : Semi-D : Terrace 5 : 3 : 2 Let the price of a semi-detached house be x : Semi-D Terrace x 2 x 3 1200000 5 1200000 5 2 3 x 1200000 x 1200000 5 5 720000 480000 A curtain of length 810m has three parts with different materials. The ratio of the lengths of PQ to QR to RS is 4 : 2 : 3. What is the length of PQ? 4 P 2 Q 810m 3 R S 4 2 P Q 810m PQ + QR + RS = 4 + 2 + 3 = 9 parts 9 parts = 810 m PQ 1 part represents = 810 m 9 = 90 m 4 parts represent = 90 x 4 = 360 m 3 R S 4 P 2 Q 810m PQ : QR : RS 4 : 3 : 2 PQ 4 810 4 2 3 4 PQ 810 9 PQ 360m 3 R S The ratio of the sides AB, BC and AC of triangle ABC are in the ratio 7 : 5 : 6. Find the length of side AB if the difference of the sides AB and BC is 4 cm C 5 6 A 7 B C 5 6 A AB-BC = 4 cm AB ─ BC = 7 ─ 5 = 2 2 parts represent = 4 cm 1 part = 2 cm AB = 7 parts × 2cm = 14 cm 7 B The ratio of the sides DE, EF and FD of triangle DEF are in the ratio 11 : 9 : 5. Find the sum of the three sides if the difference of the sides DE and DF is 18 cm F 9 5 D 11 E F 9 5 D 11 DE ─ DF = 18 cm DE ─ DF = 11 ─ 5 = 6 6 parts represent = 18 cm 1 part = 3 cm Sum of all sides = 11 + 9 + 5 =25 parts = 25 parts × 3cm = 75 cm E