Exploring Ratios and Proportions in Year 9 Mathematics
Dive into the world of ratios and proportions with Dr. J. Frost's engaging exercises and puzzles for Year 9 students. Explore concepts such as direct proportion, unitary method, and solve problems involving fish tanks, polygons, and more. Strengthen your understanding of mathematical relationships through practical examples and challenges.
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Year 9: Ratio & Proportion Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Last modified: 3rd September 2015
Starter Puzzles (You can work in pairs) My fish tank has black and yellow fish in the ratio 3:1. A fish plague, Frostitus, wipes out a third of my fish. I then restock my fish tank with just black fish, so that I have the same number of fish as before. What s the new ratio of black to yellow fish? 1 3 In a cardboard box, the ratio of cats to dogs is 3:5 and the ratio of dogs to sharks is 4:7. What is the ratio of cats to sharks? Make parts dog same in each. So cats to dogs ??:?? and cats to sharks ??:??. So ratio of cats to sharks ??:??. ? There are two regular polygons, ?and ?. The ratio of their exterior angles is 1:3. The ratio of their interior angles is 7:6. Prove that polygon ? has 30 sides. (Hint: it may be helpful to start with the exterior angle of ? being ? and working from there) ? Answer: ?:? 2 The ratio of males to females at a party is 3:5. There are twelve more females than males. How many people are at the party? ??? ? ??? ??=? ? ? ? = ?? Num sides = ??? ??= ?? ? Answer = ??
Direct Proportion Iain wants to make a Baked Alaska. He requires 24 eggs to make 700g of the pudding. How many eggs does he require to make 1250g? Method 1: Unitary Method (Find number of eggs for 1g) 700g require 24 eggs. So 1g requires 24 So 1250g requires: 24 700 1250 = 42.857 So 43 eggs. 700 eggs. ? Method 2: Ratio Method If ?:? = ?:?then ? ?=? ? 1250:700 = ?:24 1250 700= 24 ? = 24 1250 ? We say that the number of eggs and the mass of the pudding are directly proportional . There is some underlying scaling between the two things. 700= 42.857
Test Your Understanding Solve the following using both the unitary method (i.e. find quantity for one unit) and the ratio method . The mass of 16cm3 of Neoginium is 24g. What is the mass of 20cm3 of the same element? Q Unitary Method Ratio Method 16cm3 is 24g. 1cm3 is ?? 20cm3 is: ??:?? = ??:? ?? ??= ?? ? = ?? ?? ? ?? ? ? ?? ?? ??= ?? ??= ??? If you finish: There are 7 Aardvarks and 12 Buffalo in a classroom. The ratio of Aardvarks to Buffalo is ? ? + 3. What is ?? Q ? + ? ? =?? ? ? ? = ?.?
Direct Proportion Two things are directly proportional if they in the same ratio. Can you suggest variables that might be directly proportional? Speed and distance travelled (if you double your speed, you double the distance travelled). Total cost and quantity purchased. Length of steel rod and weight. ? Why is hours revised for maths exam and maths exam mark not likely to be directly proportional? If we 5 hours revision resulted in 60%, then clearly 10 hours revision wouldn t result in 120%! So while the two things are correlated , they are not directly proportional. ? Electricity cost is directly proportional to the hours of TV watched. If 2 hours are watched, a cost of 14p is incurred. If 7 hours is watched, what cost is incurred? Q 14 2 7 = 49?
Exercise 1 (see provided sheet) 5 The rates of currency exchange published in the newspapers on a certain day showed that 14 kroner could be exchanged for 210 pesos. How many pesos could be obtained for 32 kroner? 480 pesos ? If it takes 3 men 5 hours to dig 12 holes, how many hours does it take for 3 men to dig 84 holes? 35 hours ? 1 [IMC 2009 Q9] Joseph s flock has 55% more sheep than goats. What is the ratio of goats to sheep in the flock? A 9:11 B 20:31 D 5:9 E 9:20 Solution: B ? 6 At a steady speed, a car uses 15 litres of petrol to travel 164km. At the same speed, what distance could be travelled on six litres? 65.6km ? 2 C 11:20 If a 2kg bag of sugar contains 9 106 crystals, how many crystals are there in: a) 5kg? ?.?? ??? b) 8kg? ?.? ??? c) 0.03kg? ?.? ???= ??? ??? [IMC 2005 Q6] A shop advertise Buy one, get one at half price . For this offer, the average cost per item is the same as: A Two for the price of one B Three for the price of one C Three for the price of 2. D Four for the price of three E Five for the price of four Solution: D (make the original cost of each item 1 for simplicity, and go from there) ? 7 3 ? ? ? The amount of energy carried by an electric current is proportional to the number of coulombs. If five coulombs carry 10 joules of energy, how many joules are carried by 6.5 coulombs? 13 joules ? 4
Exercise 1 (see provided sheet) If it takes ? men ? hours to dig ? holes, how many holes can be dug in ? hours? ? [IMC 1997 Q18] Seven rolls weigh the same as four crumpets. Five scones weigh the same as six crumpets. If each crumpet weighs ? grams, and each roll weighs ? grams, and each scone weighs ? grams, which of the following must be true? A ? < ? < ? C ? < ? < ? E ? < ? < ? Solution: E ? 8 ?? ? [IMC 2010 Q22] Curly and Larry like to have their orange squash made to the same strength. Unfortunately, Moe has put 25ml of squash with 175ml of water in Curly s glass and 15ml of squash with 185ml of water in Larry s glass. How many millilitres of the mixture in Curly s glass must be put into Larry s glass so that they end up with drinks of the same strength? A 5 B 7 C 10 E it is not possible Solution: E [IMC 2003 Q25] The width : height ratio of television screens is changing from the traditional 4 : 3 to the widescreen 16 : 9. If a traditional screen and a widescreen have the same area, then what is the ratio widescreen width : traditional width? A 2: 3 B 3:2 D 4:3 E none of these Solution: A ? B ? < ? < ? D ? < ? < ? D 12 ? The number of coins required to cover a circular table is proportional to its area. When the radius of the table is 1m the number of coins is 100. How many coins can cover it when the radius is 2m? 400 coins ? 9 C 3: 2
Inverse Proportion Suppose Mo runs at a speed of 8m/s for second, and takes 120 seconds to finish. If he ran double the speed at 16m/s, what happens to his time? It halves! ? And what do you notice about the speed multiplied by the time in the two cases? It remains the same. ? When two quantities are inversely (or indirectly ) proportional, their product remains constant.
Example Four bricklayers can build a certain wall in ten days. How long would it take five bricklayers to build it? Q Method 1: Constant Product Method 2: Unitary Method Suppose it takes five bricklayers ? days to build the wall: 5 ? = 4 10 5? = 40 ? = 8 Find how many days one bricklayer would take! ? ? 4 bricklayers take 10 days. 1 bricklayer takes 40 days 5 bricklayers take 8 days
Test Your Understanding Eleven taps fill a tank in three hours. How long would it take to fill the tank if only six taps are working? Q If using constant product method: 11 3 = 6 ? ? =33 6= 5.5 ???? ? If using unitary method: 11 taps fill tank in 3 hours 1 tap fills tank in 33 hours 6 taps fills tank in 33 6= 5.5 hours If you finish that quickly If it takes 3 men 4 hours to dig 6 holes, how many hours does it take for 6 men to build 8 holes? Q Notice that men and hours are inversely proportional but hours and holes are directly proportional. ? 3 men 4 hours for 6 holes 6 men 2 hours for 6 holes 6 men 1/3 hours for 1 holes 6 men 8/3 hours for 8 holes (which is 2 hours 40 minutes)
Exercise 2 The length of an essay is 174 lines with an average of 14 words per line. If it is rewritten with an average of 12 words per line, how many lines will be needed? ??? A marathon runner increases their speed by 25%. What percentage does their time decrease by? 20% ? 6 1 ? Nine children share out equally the chocolates in a large tin and get eight each. If there were only six children, how many would each get? ?? ? 2 If it takes ? men ? hours to dig ? holes, how many holes can be dug by ? men in ? hours? ? men ? hours for ? holes 1 man ?? hours for ? holes ? men ?? hours for ?? holes ? men 1 hour for ??/?? holes ? men ? hours for ???/?? holes 7 A batch of bottles were packed in 25 boxes taking 12 bottles each. If the same batch had been packed in boxes taking 15 each, how many boxes would be filled? ?? ? 3 ? In a school, 33 classrooms are required if each class has 32 pupils. How many classrooms would be required if the class size has reduced to 22? ?? A fixed amount of water is filled to the top of a cylindrical container. If the height of the cylinder doubles, but the water still fills to the top, what happens to the radius? (Note that ?????? = ???? ?? ?????? ??? ?) It becomes ? times smaller. ? 4 8 ? If it takes 6 men 25 hours to dig 5 holes, how many hours does it take to dig 5 holes with 10 men? ?? ????? ? 5
