**Average Rates**

Published on ginnyent.net on 12th march, 2019

Photo: www. kiplinger.com

I want to take a little time to apologise for last week, for not posting on the Mathematics Corner Tuesday and Thursday. Please pardon me; it was because of a circumstance beyond my control.

This week we will be talking about Average Rates.

The meaning of the word ‘average’ in mathematics – it means a quantity got by dividing the sum of several quantities by the number of the given quantities. Is the definition ambiguous? Let’s make it clearer.

Consider the following quantities: 3, 7, 9, 10, 15, 21, 25; if you’re asked to find the average of these quantities, what do you do?

First: add the quantities together

Second: divide the sum by the number of the given quantities

So we have: 3 + 7+ 9+ 10+ 15 + 21 + 25= 90

We have done the first step.

The second step says; divide the sum by the number of the given quantities.

So from the first step, our sum is 90. The number of the given quantities is 6(count the quantities given).

Therefore the average is 90/6

= 15

Now talking about **Average Rates** or **Average of Rates**, we are going to find the average of rates like 3km/h and 6km/hr, $50/day and $150/day etc.

__Example 1__

From Monday to Wednesday, Mr. Afolabi made a profit at the rate of $500 per day. Between Thursday and Saturday, he made a profit at the rate of $700 per day. Calculate his average daily profit for the six days.

**Solution**__ __

We need to find the total profit he made from Monday to Wednesday and from Thursday to Saturday.

From Monday to Wednesday, Mr. Afolabi made: $500 x 3 =$1500

From Thursday to Saturday, Mr. Afolabi made : $700 x 3 = $2100

Now, we will add the profits together: $1500 + $2100 = $3600

Remember our second step says; divide the sum by the number of quantities. Here, we have six quantities actually because, Mr. Afolabi makes profit daily. So we are going to divide by 6.

Therefore the average rate is: $3600/6

= $600 (Understood?)

**Example 2**

Paul travelled from kainji to Bauchi, a distance of 180km last Thursday. He boarded a bus which covered 9km/hr and from Kainji to Gusau where he alighted. He walked 2km at the speed of 10metres/hr to get to a park where he boarded another bus from Gusau to Bauchi, covering a distance of 100km at 5km/hr.

- Find the average speed for Paul’s journey.
- Find the approximate number of days it took him to arrive Bauchi ?

**Solution**

- We will take three steps here:

First, we have to find the total distance covered.

Second, we have to find the time for each part of the journey

Third, we will divide the total distance by the total time taken.

Total distance covered = 180 + 2 + 100

= 282km

Time for the first part of the journey = 180/9 (distance covered/speed)

= 20 hours

Time taken for the second part (the walk): here notice the difference in units, km and metres. So we need to have the quantities in the same unit.

1000metres = 1km

100metres = __(1 x100__)

1000

= 100/1000

= 0.1km

Time taken for the second part of the journey = 2/0.1

= 20 hours

Time taken for the third part of the journey (from Gusau to Bauchi) = 100km/5

= 20 hours

Total time taken for the whole journey = 20 + 20+ 20

= 60 hours

Average speed = __Total Distance __

Total Time Taken

= 282km/60 hours

= 4.7km/hour

- To calculate how many days he made the journey,

24 hours make one day;

So in 60 hours, we have 24 + 24 + 12

= 2 days and half

= 3 days approximately.

__Exercise __

- On a journey, a motorist travels the first 50km in an hour and half, then the next 34km in 25 min and the last 7km in 5 min. what is the average speed for the whole journey?
- A factory employs 50 workers. 40 earn $90/hour and 10 earn $120 /hour. What is the average hourly rate of pay?
- A lorry travelled 84km between two towns. The first 60km of road was un-tarred and the average speed over this part was 30km/hr. If the average speed for the whole journey was 36km/hr, calculate the average speed over the good part of the road.

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