When one is starting on a Math topic, it is important to figure out the scope of the topic and the inferences specific to that topic.
It is also helpful to link that topic to other similar topics. For example Rate problems are close cousins to Speed problems. Both employ time concepts. Though each have their own specific inferences.
Identifying the major categories of problem sum in any topic is critical for a detail understanding of that topic. It forms a boundary around that topic and helps the child to explore all areas within that topic and flush out weaknesses in any specific category.
The diagram below illustrates the 4 main categories for Rates.
Each major category introduces new concepts.
Within each major category, there are sub categories. The sub categories do not require the student to understand new concepts, but the relationships are presented in a new way, which might be confusing to the student initially.
Pure Rates are problems that involve Jobs.
Here is an example of Pure Rate problem.
J takes 6 hours to paint a house. If D helps him, they would take 4 hours. How long will D take to paint the house himself?
(Ans: 12 hrs)
In this category of problems, one has to understand that rates are additive (ie. Rates can be added together). But only if it is specifically mentioned that more than 1 person is working on a job at the same time. If it is not mentioned, then we cannot assume that the Rates are additive. In the above problem, since D helps J paint the house, then we can assume that there is a combined Rate concept. The combined Rate concept is simply the addition of 2 or more rate values. ie. Rate of J adds to the Rate of D will give the combine Rate.
In the below question, it is not stated that Desmond and Philip work on the renovation job together. We thus cannot assume that the rates here are additive.
Desmond takes 9 days to renovate 1 kitchen while Philip takes 21 days to renovate the same kitchen. If Philip starts renovating the kitchen first and leaves the rest to be completed by Desmond, they will take 13 days in total to complete renovating the kitchen. How many days does Desmond spend on the job?
(Ans: 6 days)
Water rates are problems that involve water and taps. Below is an example of Water Rate problem.
At 9am, Mr Fernandez used 2 taps to fill up a tank. The first tap could fill the tank in 4 hours. The second tap could fill the tank in 3 hours. An hour after both taps were turned on, the second tap were faulty and stopped working. Mr Fernandez accidentally opened the 3rd tap which could drain a full tank completely in 2 hours. Instead of being filled, the tank was being emptied. How long did it take for the tank to be completely empty ?
(Ans: 2 hours 20 mins)
Water rate problems are characterized by additive rates and negative rates, something that is not found in Pure Rate problems. When 2 taps are turned on, you add up the rates of the 2 taps to get the combine rate of both taps. Positive rate values are when the tap is filling the tank with water. Negative rate values are when the tap is draining water away. So if a tap is filling up the tank at a rate of 2 liters/min and another tap is draining water away at at rate of 1 liter/min, then the combine rate is (2 – 1) liter /min. If the draining tap has a rate of 3 liters/min, then the combine rate is (2 -3) = -1 liter/min, which means that on the whole, water is being drained away from the tank.
Higher order Water Rate problems will involve Clock time. ie. At 9am, Tap A is turned on, at 9.05am, Tap B which drains water away is turned, at 9.10am, Tap C is turned on and Tap A is turned off. It can get pretty complex with the introduction of clock time. One way to visualize the complex logic is to draw a timeline, to understand at which duration, what taps are turned on or off. Typically for time concepts, timeline diagrams are drawn, as in Speed problems. In problem sums involving time, the time duration is required. Clock time is basically an indirect way of telling us the time duration.
Volume rates requires the student to understand how volume (length x breadth x height) interacts with Rates. On top of Rate concepts, students has to understand Volume concepts. A common problem in Volume Rates involves finding the height of the water in the tank. Typical approach to this is to convert the rate of water flow to rate of height increase/decrease. However, the conversion step is not necessary if the student does not mind working with larger numbers.
Tap X flows at a rate of 2100 ml/min while Tap Y flows at a rate of 2500ml/min. Both taps were turned on at the same time to fill a tank with dimensions 50 cm x 40 cm x 30 cm. After 5 minutes, the plug at the bottom of the tank is removed, with the two taps still running. If the water is drained at a rate of 600 ml/min, what is the water level 2 minutes after the plug is removed?
The concept of clock time, additive rates and negative rates are also introduced here as well.
The figure below shows 2 rectangular tanks, X and Y. Tank X contained 5184 liters of water and Tank Y contained 1755 liters of water.
Tap A and Tap B were turned on at 8.00 am. Water flowed out of Tap A at 34 liters/min and out of Tap B at 7 liters/min. When both tanks had the same volume of water, both taps were turned off.
(a) At what time were the taps turned off? (Ans: 10:07am)
(b) What was the volume of water in each tank when the taps were turned off?
(Ans: 866 liters)
You don’t normally see Candle Rates problems around in assessment books. It is the least common type of Rate problems. Students should first work on Speed problems and be familiar with Common Time and Common Distance before working on Candle problems. The concepts learned in Speed can be reused here.
Two candles of the same height are lit at the same time. The first candle takes 5h to burn completely. The second candle takes 4h to burn completely. If each candle burns at a constant rate, how long does it take, in hours, for the height of the first candle to be four times that of the second candle?
(Ans: 3 hours 45 mins)
Pure Rates Again
Within Pure Rate Problems there are 4 sub categories:
a) Individual Rates:
Individual rate problems are characterized by having no Combine Rate concepts. One cannot add the rates together simply because the problem sum did not state that individuals are working together.
When Caleb spent 2 days and Ethan spent 3 days assembling computers, they produced 42 sets of PCs. When Caleb spent 4 days and Ethan spent 2 days assembling computers, they produced 44 sets of PCs. How long will each boy require to assemble 30 sets of computers?
(Ans: Ethan: 3 days, Caleb: 5 days)
b) Whole Combinations
Whole combinations are characterized by problem sums where the individuals work together to finish off an entire job. It is common that the individuals will work alone for a period of time, and then work together for another period of time to finish the entire job. Combine Rate concept is involved when individuals work together.
Ahmad and Halim together took 5 days to paint their house. If Ahmad and Halim work together for 2 days, followed by Ahmad working alone for 8 days, Halim will take 1 more day to complete the remaining work. How long will Ahmad take to paint the house all by himself?
(Ans: 17.5 days)
c) Partial Combinations
Partial combinations are characterized by problem sums where the individuals finishes off part of the job, i.e only a partial completion of the job is done.
In the example below, Allan and Bob only finishes 5/12 of the job.
If Allan and Bob work together, they can complete a job in 12 days. If they work together for 3 days, followed by Bob who worked alone for the next 5 days, they can finish 5/12 of the job. How many days will each of them need to complete the job if they were to work alone?
(Ans: Bob 30 days, Alan 20 days)
d) Multi Combinations
Multi combination problem sums involves multiple equations. The rate concepts introduced here are basic concepts, but the difficulty is in resolving the multiple equations to find the value of a specific variable.
John and Rauf take 4 days to build a model train. Rauf and Sean take 6 days to build the same train while John and Sean take only 3 days. How long would the three of them take to build the same model train?
(Ans: 2 2/3 days)
Abel and Ben can build a model aircraft together in 4 days. Ben and Calvin can do the same job together in 1 1/2 times as many days. Abel and Calvin can complete the same job together in twice the number of days Ben and Calvin can do together. If the three boys decide to work together, how long will they take to complete their job?
(Answer: 4 Days)