08-17-2017, 12:44 AM
Objectives:
Help students to discover what makes the Percentages important in almost all the tests.
Help students to learn how to deal with different real-time applications of percentages.
Help students to understand how to go about the study material and what to do before coming to
the next lecture.
Help students to make use of LOGIC in place of conventional methods.
Step 1: Fundamentals - 1
1. Change of base
2. Successive percentage change
Step 2: Fundamentals - 2
3. Simple and compound Interest
4. Interest on interest
5. Formulae for SI and CI
6. Non-annual compounding
Step 3: Class Exercise
Give them sufficient time and discuss few problems from class exercise at the end of the lecture. Tell them
to get the doubts solved with the faculty available at the center before the next lecture.
NOTE: Make necessary announcements.
All the things that are to be discussed in the lecture have been summarized in the
following few pages. Read them carefully and don t miss anything.
A brief Introduction (5-10 mins)
2. Change of base (15 mins)
Start with the simplest problem: If A is 20% more than B, by what percent is B less than A? Do the
same problem if A was 37.5% more than B. Explain why it is important to work with fraction equivalents
to save time & calculations.
Explain that the above simple concept is used extensively even in other topics as follows:
a. if speed increases by 10% over the same distance, by what percent does time decrease?
b. If length of a rectangle decreases by 12.5%, by what percent should its breadth be increased to
maintain the same area?
c. If the volume of a milk and water solution is increased by 25% by pouring just water, by what
percentage does the concentration of milk reduce?
d. If prices decrease by 16.666%, how much percentage can a consumer consume more for the
same amount?
[Q. 2, 6, 8 and 9]
Explain that the above can be used in any relation where M N = constant.
If prices of apples increase by 25%, I am able to purchase 4 apples less in Rs.80. What was the
original price of one apple? Explain this question orally in the following two methods: Since prices
increased by 25%, I am able to consume 20% less. Thus 20% of original consumption = 4 apples.
Alternately, to maintain consumption i.e. to purchase 4 more apples, I would need 25% of 80 = 16 Rs
more. Thus increased price of apple = Rs. 4 per apple. Now find original price.
3. Successive Percent changes (10 mins)
Take a simple case of two successive increases say 10% and 20% and then explain why the total
increase is 32%. State that the net increase of an (a%) and a (b%) change is a (a + b + ab/100%)
change. Do mention that the formula works fine with any changes viz. increase or decreases. Ask
student to find the net percentage change of a x% increase and a x% decrease and also for a 10%
increase and a 9.0909% decrease. Probably the last example can be more easily solved with multiplying
factors. Thus one can use Multiplying factors or a + b + ab/100 interchangeably.
Successive percentage change is also useful in any relation of the type C = A B. If there is a (a%)
change in A and a (b%) change in B, then C changes by (a + b + ab/100%). This has also applications
in Data Interpretation. Thus if market share grows by 20% and even if the total market size declines by
10%, the sales grows by 1.2 0.9 = 1.08 i.e. 8% as Sales = Market size Market Share.
The same relation appears many times in geometry. Thus if any quadrilateral has all its sides increasing
by 10%, the area increases by 21% as area is proportional to square of linear dimensions. If sides of
a cuboid increase by 20%, volume increases by 72.8% and surface area increases by 44%.
[Q. 1 and 3]
CAT 2003 (Leaked): Let A and B be two solid spheres such that the surface area of B is 300% higher
than the surface area of A. The volume of A is found to be k% lower than the volume of B. The value of
k must be
a. 85.5 b. 92.5 c. 90.5 d. 87.5
4. Simple and Compound Interest (10 mins)
Quickly ask students for the basic difference between SI and CI i.e. the amount at end of year acts as
principal for next year in CI. Explain this difference using the following table :
For P = 1000 and r = 10%,
(draw this table such that you still have additional space on the board to write)
Year Simple Interest Compound Interest
Principal Interest Amount Principal Interest Amount
1st 1000 100 1100 1000 100 1100
2nd 1000 100 1200 1100 110 1210
3rd 1000 100 1300 1210 121 1331
Points to be highlighted in this table:
a. Simple Interest earned in any year is always the same
b. The amount with SI increases in a linear fashion i.e. by a fixed amount every year. Thus the graph of
amount and years will be a straight line.
c. The SI and CI is same for the first year
d. CI earned in a year keeps increasing every year.
5. Interest on Interest (10 mins)
With the above table still present on the board, explain point number d of above in more details.
Explain that in any year the CI earned is higher because whatever was earned in the previous year
would surely be earned and additionally the interest earned in previous year will also be added to the
principal and start earning interest. Make sure this concept of Interest on Interest is understood by
everyone. Take a simple problem : If CI earned in nth year is 800 and in n+1th year is 864, what is the
rate of interest?
Next, compare the first two years at SI and CI. For same principal and rate of interest, in the first two
years, the total CI earned is higher than the total SI earned by an amount equal to the interest on first
years interest. Take simple problems like :
If total SI earned in first two years is Rs. 800 and total CI earned in first two years is Rs. 864, what is
the rate of interest?
Explain the following:
For same principal and rate of interest,
Year Simple
Interest
Compound
Interest
1st X X
2nd X X + r X
Total 2X 2X + r X
Thus explain the difference is r% of X and also that the ratio of CI of first two years to SI of first two
years is (2 + r) : 2
Take another simple example : If the difference between CI and SI of first two years @ 10% is Rs. 500,
what is the principal invested.
If the ratio of SI to CI earned in first two years is 24 : 25, what is the rate of interest?
6. Formulae for SI and CI (10 mins)
Explain the formulae :
n P r n r
SI At CI, A P 1
100 100
= = +
Explain that for CI, the formula is of Amount and for SI the formula is of Interest.
In the formulae of SI, make sure to highlight the fact that SI is directly proportional to P, r and n. Thus
if an amount becomes 3 times in 7 years, in how many years will it become 9 times? Please explain
to all students who have answered 21 years that the Amount is not directly proportional to the number
of years but the Simple Interest is. In this case the SI earned is 2P in 7 years and hence to earn a SI
of 8P it will take 28 years. However had it been the case of CI, the amount would have become 9 times
in 14 years itself because in every 7 years the amount triples, thus in next 7 years 3P will become 9P.
Also spend some time explaining that for a r% increase the Multiplying factor is (1 + r/100). Thus for 2
years at CI, the amount is nothing but P MF MF and thus CI is in essence a case of successive
percentage changes.
[Q. 4, 5 and 10]