 ## Ratio and Proportion

### Ratio and Proportion

Ratio is used to measure the quantity of an object in terms of the other object(s).

In the olden days you may have to trade in the barter sytem.

Or, you are going on a holiday and you need to exchange your currency to another country’s currency.

Or even when you are going to make a cake and need to mix a variety of ingredients.

These are some of the examples of which a good basic knowledge on mathematical ratio might really help you.

At least you wouldn’t want your cake to taste awful, would you? =) .

Ratio can be expressed by either a colon or as a fraction between the two quantities you want to compare.

Let’s see the example below:

$$\hspace{2em} A \enspace : \enspace B \enspace = \enspace 2 \enspace : \enspace 5$$

$$\hspace{4.2em} {\large \frac{A}{B} \enspace= \enspace \frac{2}{5} }$$

The ratio of A to B is “2 to 5”.

Meanwhile in proportion, to increase or decrease the ratio by the same amount we multiply the ratio by the same number.

For example, if Andy has a bag consists of ten candies, how many candies would there be in 5 bags? Therefore, Or we can calculate the proportion of the ratio between the bags and candies, There are 50 candies in 5 bags.

Just remember, there are a few things to keep in mind:

– To simplify a ratio expression, you can use the HCF (highest common factor) to divide both numbers in the ratio.

– More often than not, it is generally helpful to sketch the diagram and solve the questions by using the box method/unitary method/the bar model.

– Sometimes it is easier to solve the problem by writing the ratio in fractions.

– If you need to compare more than 2 quantities, find the LCM (least common multiple) of the same quantity that is being compared.

Try some of the examples below and if you need any help, just look at the solution I have written. Cheers ! =) .
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EXAMPLE:

$${\small 1.\enspace}$$ What is the missing number in the box?
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$$\hspace{2.5em} 3 \enspace : \enspace 5 \enspace = \enspace {\large \square } \enspace : \enspace 30$$

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$${\small 2.\enspace}$$ Jennifer has 18 stamps. She has 10 more stamps than Isabelle. Find the ratio of the number of stamps Jennifer has to the number of stamps Isabelle has.

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$${\small 3.\enspace}$$ The ratio of the number of men to the number of women working in a factory is 5 : 2. There are 105 workers in the factory. How many men are working in the factory?

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$${\small 4.\enspace}$$ At a party, there were 8 women to every 15 men. 35 more men than women attended the party. How many women were there at the party?

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$${\small 5.\enspace}$$ The ratio of A : B is 2 : 5. The ratio of B : C is 7 : 4. Find the ratio of A : B : C.

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$${\small 6.\enspace}$$ The ratio of the number of boys to the number of girls at a workshop was 5 : 6. After 22 boys left the workshop, the ratio of the number of boys to the number of girls who remained became 2 : 3. Find the total number of students who attended the workshop at first.

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$${\small 7.\enspace}$$ The ratio of Uncle Gary’s age to his son’s age now is 5 : 2. Given that the ratio of the their ages 10 years ago is 5 : 1, find Uncle Gary’s age now.

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$${\small 8.\enspace}$$ The ratio of the number of tennis balls in Basket A to the number of tennis balls in Basket B was 5 : 4. When 8 tennis balls were taken out from Basket A and placed into Basket B, there was an equal number of tennis balls in each basket. How many tennis balls were there in Basket A at first?

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$${\small 9.\enspace}$$ At a swimming carnival, the ratio of the number of swimmers to the number of non-swimmers is 8 : 9. The ratio of the number of female swimmers to the number of male swimmers is 4 : 7. If $${\small {\large \frac{2}{3}} }$$ of the non-swimmers are females and there are 64 female swimmers, how many male and female non-swimmers are there at the swimming carnival?

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$${\small 10.\enspace}$$ Elisa, John and Wilson donated a sum of money to the school building fund. Elisa donated $${\small {\large \frac{3}{10}} }$$ of the money. The remaining was donated by John and Wilson in the ratio 5 : 9. Elisa donated $900 less than Wilson. $${\small\hspace{1.2em}(\textrm{a}).\hspace{0.8em}}$$ How much did John donate? $${\small\hspace{1.2em}(\textrm{b}).\hspace{0.8em}}$$ For every one dollar donated by Elisa, how much did Wilson donate? $$\\[1pt]$$ PRACTICE MORE WITH THESE QUESTIONS BELOW! $${\small 1.\enspace}$$ The ratio of the number sweets Alfred had to the number of sweets Brandon had was 3 : 8. After each of them received 45 sweets, the ratio of the number sweets Alfred had to the number of sweets Brandon had became 3 : 5. How many sweets did they have altogether at first? $${\small 2. \enspace}$$ The length and breadth of a rectangle are in the ratio 7 : 4. Given that the perimeter of the rectangle is 352 cm, find its area. $${\small 3. \enspace}$$ John’s present age to Andrew’s present age is 3 : 1. In 6 years’ time, the ratio of John’s age to Andrew’s age will be 5 : 2. What is John’s present age? $${\small 4. \enspace}$$ Grace and Alice have some beads in the ratio 9 : 5. Grace gave 28 of her beads to her sister and Alice bought another 32 beads. At the end, Grace and Alice have the same number of beads. How many beads did they have altogether at first? $${\small 5. \enspace}$$ Greg went shopping with$360. He bought some calculators and shirts and had $8.50 left. Each shirt cost$15.50 more than a calculator. He bought 3 fewer calculators than shirts. How many shirts and calculators did he buy if each calculator cost $17.50? $${\small 6. \enspace}$$ When Belinda purchased 6 kg of rice, she was short of$6. When she purchased 8 kg of the same type of rice, she was short of $14. How much money did she have? $${\small 7. \enspace}$$ If Liming gives Dave$8, both of them will have the same amount of money. If Florence adds \$14 to Liming’s original sum of money, the ratio of Liming’s money to Dave’s original sum of money will be 5 : 2. How much has Liming at first?

$${\small 8. \enspace}$$ Peter, Cecilia and Tony have a certain number of books. If Peter gives 6 books to Cecilia, he will have half as many books as she. If Tony gives 10 books to Cecilia, both of them will have the same number of books. If Cecilia has 30 books at first, how many books do Peter and Tony have altogether at the beginning?

$${\small 9. \enspace}$$ Joanne and Bernard had a certain number of stamps each. If Joanne gave 90 stamps to Bernard, they would have the same number of stamps. If Bernard gave 10 stamps to Joanne, she would have 5 times as many stamps as Bernard. How many stamsps did Joanne have at first?

$${\small 10.\enspace}$$ A group of boys played only two games. $${\small {\large \frac{1}{3}} }$$ of them played basketball and $${\small {\large \frac{1}{4}} }$$ of them played football. $${\small {\large \frac{1}{6}} }$$ of them played both games and 56 boys did not play any games at all. How many boys were there?
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As always, if you have any particular questions to discuss, leave it in the comment section below. Cheers =) . ## Speed

### Speed

This is a continuation of the Ratio topic that we have previously learned. The distance you are covering is proportional to the amount of the time taken.

The more time you spend on a journey, the more distance you will be covering. But some are slower and some are faster. To quantify this we use speed.

Speed is the rate of the distance travelled. It is the amount of distance covered over the time taken, simply a ratio of distance and time. It tells you how fast you are moving.

$${\large\boxed{s=\frac{d}{t}}}$$

where:
s $$=$$ speed
d $$=$$ distance
t $$=$$ time

Another important concept here is the average speed. It is the total distance travelled over the total time taken for the entire journey. Students sometimes mistakenly calculate the average speed by taking the average of the speeds.

As an example, let’s say you travelled the first part of a journey by covering 50 km in an hour and then completed the second part of the journey by covering 60 km in two hours. What will be the average speed of the journey?
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The wrong calculation is typically done this way:
Step 1:$$\enspace$$ Find the speed of the first part of the journey.

$$\\[7pt]\hspace{5em}{ s }_{ 1 } = \frac{50\, \mathrm{km}}{1\, \mathrm{h}}$$
$$\hspace{6.2em} =$$ 50 km/h

Step 2:$$\enspace$$ Find the speed of the second part of the journey.

$$\\[7pt]\hspace{5em}{ s }_{ 2 } = \frac{60\, \mathrm{km}}{2\, \mathrm{h}}$$
$$\hspace{6.2em} =$$ 30 km/h

Step 3:$$\enspace$$ Find the average speed by taking the average of both speeds.

$$\\[7pt]\hspace{5em}{ s }_{ average } = \frac{50\, \mathrm{km/h}\;+\;30\, \mathrm{km/h}}{2}$$
$$\hspace{8.2em} =$$ 40 km/h

This is wrong as the average speed must be found by first calculating the total distance traveled over the time taken!
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The correct way of solving this problem is:
Step 1:$$\enspace$$ Find the total distance of the journey.

$$\hspace{5em} {\small d = 50\, \mathrm{km}\; + \; 60\, \mathrm{km}}$$
$$\hspace{5.7em} =$$ 110 km

Step 2:$$\enspace$$ Find the total time taken of the journey.

$$\hspace{5em} {\small t = 1\, \mathrm{h}\; + \; 2\, \mathrm{h}}$$
$$\hspace{5.5em} =$$ 3h

Step 3:$$\enspace$$ Find the average speed by taking the ratio of total distance to the total time taken.

$$\\[10pt]\hspace{5em} { s }_{ average } = \frac {50\; \mathrm{km}\;+ \; 60\; \mathrm{km}}{1\; \mathrm{h}\; + \; 2 \; \mathrm{h}}$$
$$\\[10pt]\hspace{8.2em} =\frac{110\; \mathrm{km}}{3\; \mathrm{h}}$$
$$\hspace{8.2em} \approx$$ 36.67 km/h

EXAMPLE:

$${\small 1.\enspace}$$ A train, 600 m long, enters an underground tunnel which is 5.4 km long at a speed of 120 km/h. How long will it take before the end of the train emerges from the tunnel? Give your answer in minutes. $$\\[1pt]$$
$${\small 2.\enspace}$$ Cecilia and Janet ran in a race. When Janet completed the run in 30 minutes, Cecilia had only run $$\frac{5}{9}$$ of the distance. Cecilia’s average speed for the race was 60 m/min less than Janet’s.
$${\small\hspace{1.2em}\left(a\right).\enspace}$$ Find the distance of the race.
$${\small\hspace{1.2em}\left(b\right).\enspace}$$ What was Cecilia’s speed in m/min?

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$${\small 3.\enspace}$$ Raju set off from his house at 1 p.m. and travelled at a uniform speed of 75 km/h towards Gopal’s house. He increased his speed to 90 km/h at 3 p.m. and reached Gopal’s house at 6 p.m.
$${\small\hspace{1.2em}\left(a\right).\enspace}$$ Find the distance between Raju’s and Gopal’s houses.
$${\small\hspace{1.2em}\left(b\right).\enspace}$$ Find the average speed of the whole journey.

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$${\small 4.\enspace}$$ At 10.00 a.m., Mr. Koh’s car passed a certain point, A, travelling at an average speed of 80 km/h. At 11.30 a.m., Mr. Bob’s car started off from point A at an average speed of 100 km/h in pursuit of Mr. Koh’s car.
$${\small\hspace{1.2em}\left(a\right).\enspace}$$ At what time did Mr. Bob’s car overtake Mr. Koh’s car?
$${\small\hspace{1.2em}\left(b\right).\enspace}$$ How far was Mr. Bob’s car from point A when it overtook Mr. Koh’s car?

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$${\small 5.\enspace}$$ Alicia left Town X and drove to Town Z. After travelling $$\frac{1}{4}$$ of the journey at an average speed of 72 km/h for 40 minutes, she stopped at Town Y to have a break for 20 minutes. Then she carried on with the journey at an average speed of 80 km/h. She reached Town Z at 3.45 p.m.
$${\small\hspace{1.2em}\left(a\right).\enspace}$$ What was the distance between Town Y and Town Z?
$${\small\hspace{1.2em}\left(b\right).\enspace}$$ What was the time when Alicia left Town X?

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$${\small 6.\enspace}$$ A car and a van were travelling from Town A to Town B which were 480 km apart. The van left Town A at 9.30 a.m. and drove towards Town B at an average speed of 60 km/h. The car set off 2 hours later than the van but it arrived at Town B 1 hour earlier.
$${\small\hspace{1.2em}\left(a\right).\enspace}$$ At what time did the car arrive at Town B?
$${\small\hspace{1.2em}\left(b\right).\enspace}$$ How much faster was the car travelling than the van per hour?

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$${\small 7.\enspace}$$ Tom and Jerry started travelling from Point X on a road but in the opposite direction. After 2 hours, they were 292 km apart. Jerry’s average speed was 10 km/h slower than Tom’s. What was Jerry’s average speed?

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$${\small 8.\enspace}$$ Louis and Edmund left Town B at the same time, heading in the opposite direction. Louis headed for Town C while Edmund left for Town A. The speed of Edmund was 12 km/h faster than that of Louis. After 30 minutes, Louis had completed $$\frac{2}{3}$$ of his journey while Edmund had completed $$\frac{1}{4}$$ of his journey. They were also 90 km apart.
$${\small\hspace{1.2em}\left(a\right).\enspace}$$ Calculate the speed of Louis.
$${\small\hspace{1.2em}\left(b\right).\enspace}$$ How far was Edmund from Town A when Louis reached his destination?

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SOME EXTRA PRACTICE WOULDN’T HURT!  🙂

$${\small 1.\enspace}$$ A man jogs around his estate for 45 minutes at 8 km/h. Find the distance the man jogged.

$${\small 2.\enspace}$$ Dennis rowed a boat at a speed of 18 m/min. After every 3 minutes of rowing, he rested for 2 minutes. At this speed, how long would he take to reach his destination which is at 126 m ahead?

$${\small 3.\enspace}$$ A train took 8 hours to travel from Town X to Town Y at an average speed of 72 km/h. An express train took only 6 hours for the same journey. What was the average speed of the express train?

$${\small 4.\enspace}$$ Belinda travelled at 80 km/h for the first $$\frac{4}{9}$$ of her journey. She then completed the remaining journey in 40 minutes at 90 km/h. How long did she take to complete the whole journey? Express your answer in hours and minutes.

$${\small 5.\enspace}$$ Richard and Karen had a car race. Richard drove at an average speed of 56 km/h and Karen drove at an average speed of 77 km/h. If Richard started 1 h 30 min earlier than Karen, how long would she take to catch up with Richard?

$${\small 6.\enspace}$$ Macy travelled at 72 km/h for $$\frac{5}{9}$$ of her journey. She took a 48 min break and continued travelling another 1.5 hours of the remaining journey at 96 km/h.
$${\small\hspace{1.2em}\left(a\right).\enspace}$$ Find the distance of her journey.
$${\small\hspace{1.2em}\left(b\right).\enspace}$$ What was the average speed of her journey?

$${\small 7.\enspace}$$ Mr. Tan left Town A and reached Town B by car in 6 hours. Cecilia left Town A later and was 60 km from Town B when Mr. Tan reached Town B. If the distance between Town A and Town B was 420 km and the average travelling speed of Cecilia was 20 km/h faster than Mr. Tan,
$${\small\hspace{1.2em}\left(a\right).\enspace}$$ how much later did Cecilia leave Town A than Mr. Tan?
$${\small\hspace{1.2em}\left(b\right).\enspace}$$ how many more minutes did Cecilia take to travel to reach Town B than Mr. Tan?

$${\small 8.\enspace}$$ Amber and Gina are sisters staying together. At 2.30 p.m., Amber travelled from her home to a jetty at a speed of 54 km/h. After some time, Gina left her home for the jetty travelling at a speed of 72 km/h. At 5.10 p.m., both Gina and Amber reached the jetty.
$${\small\hspace{1.2em}\left(a\right).\enspace}$$ At what time did Gina leave her home?
$${\small\hspace{1.2em}\left(b\right).\enspace}$$ Find the distance Amber drove from her home to the jetty.

$${\small 9.\enspace}$$ Mr. Chia, on his way from his house to Town Halong, after covering $$\frac{3}{8}$$ of his journey, passed a lorry which was travelling at an average speed of 80 km/h. When Mr. Chia reached his destination 4 hours later, the lorry was still 60 km away from Town Halong. If the lorry had started from Town Bay which was 48 km away from Mr. Chia’s house, how long did the lorry take to travel from Town Bay to Town Halong?

$${\small 10.\enspace}$$ Two boys were competing in a race around their school’s 400 m track. From the same starting point, Mathew ran at a speed of 1.2 m/s and Thomas ran at a speed of 1.6 m/s. How long, in seconds, did Thomas take to pass Mathew twice?
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As always, leave your comments for any questions or discussion. Cheers =) .