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<body bgcolor="aqua"><center><h2>True Discount</h2></center>
<pre>
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<strong>Formulae:</strong>
Let rate=R%per annum, Time= T years
1.Present worth (PW) = (100*Amount)/(100+(R*T))
= (100* TrueDiscount)/(R*T)
2.TrueDiscount (TD) = (P.W*R*T)/100
= (Amount*R*T)/(100+(R*T))
3.Sum =(SimpeInterest*TrueDiscount)/(SimpleInterest-TrueDiscount)
4.SimpleInterest-TrueDiscount=SimpeInterest on TrueDiscount
5.When the sum is put at CompoundInterest,then
PresentWorth=Amount/(1+(R/100))^T
<b>GeneralConcept:</b>
Suppose a man has to pay Rs.156 after 4 years
and the rate of interest is 14%per annum
Clearly ,Rs.100 at 14% will amount to Rs156 in 4 years
So,the payment of Rs.100 now wil cear off the debt
of Rs156 due 4 years hence
We say that
Sum due = Rs156 due 4 years hence
Present Worth =Rs100
True Discount=Rs.156-Rs100=Rs56
=Sumdue-PW
We define
TrueDiscount= Interest on Present Worth(PW)
Amount = PresentWorth+TrueDiscount
1.Find the present worth of Rs.930 due 3 years hence at 8% per
annum.Aso find the discount?
Sol: Amount=RS.930,Time=3years,Rate=8%
TrueDiscount = (Presentworth*Time*Rate)/100
Presentworth = (Amount*100) /(100+(R*T))
Presentworth = (930*100)/(100+(8*3)
=Rs.750
TrueDiscount = (930*3*8)/(100+(8*3)) =Rs.180 (or)
TrueDiscount = Amount-Presentworth
=Rs.930-Rs.750
=Rs.180
2.The truediscount on a bill due 9 months hence at 12% per
annum is Rs540.Find the amount of the bill and its presentworth?
Sol: Time=9months=9/12years=3/4years
Rate=12%
TrueDiscount=Rs540
TrueDiscount=(Amount*100)/(100+(R*T))
=>Amount=(TrueDiscount*(100+(R*T))) /100
=(540*(100+12*(3/4))/100
=Rs.6540
Presentworth=Amount-TrueDiscount
=Rs.6540-Rs540
=Rs.6000
3.The TrueDiscount on a certain sum of money due 3 years hence
is Rs.250 and SimpeInterest on the same sum for same time and
same rate is Rs375 find sum and rate%?
Sol: Time=3 years
Truediscount=Rs.250
SimpeInterest=Rs375
Sum = (SimpeInterest*TrueDiscount)/(SimpeInterest-TrueDiscount)
= (375*250)/(375-250)
=Rs.750
SimpeInterest=(Principle*Time*Rate)/100
375 = ( 750*3R)/100
R=50/3%
=> Rate=162/3%
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4.The difference between SimpeInterest and TrueDiscount on a
certain sum of money for 6 months at 121/2% per annum is Rs25.
Find the sum.
Sol: Let amount be Rs.x
SimpleInterest=(Amount*T*R)/100
TrueDiscount=(Amount*T*R)/(100+(R*T))
SI-T.D=Rs25
=>((x*6/12*25/2)/100)-((x*6/12*25/2)/(100+(6/12*25/2))=25
=>x=Rs.6800
5.The Present worth of Rs.2310 due 21/2yrs henceThe interest
being 15% per annum is
sol: Amount=Rs2310,Time=21/2yrs,Rate=15%
Presentworth = (Amount*100) /(100+(R*T))
=2310*100/(100+15*5/2)
=Rs1680.
<b>Medium Problems</b>
1.A bill falls due in 1 year.The creditor agrees to accept
immediate payment of the half and to defer the payment of
the other half for 2 years .By this arrangement he gains
Rs.40.what is the amount of bill,if the money be worth 121/2%?
sol: Let the amount be Rs.x
If the bill falls due for 1 yr,
Truediscount=(Amount*100)/(100+R*T)
=(x*100)/(100+(25/2*1)
If Accepting immediate payment of the half and to
defer the payment of the other half for 2 years,
Truediscount=(Amount*100)/(100+R*T)
=[x/2+((x/2*100)/(100+(25/2*2))]
x/2 for immediate payment,
((x/2*100)/(100+(25/2*2)) for paying after 2 yrs
He gains Rs40
=>[x/2+((x/2*100)/(100+(25/2*2))] a€“ (x*100)/(100+(25/2*1) = Rs40
=> x/2+2x/5-8x/9=40
=>x=Rs.3600
Amount of the bill=Rs.3600
2.If the truediscount on a sum due 2yrs hence at 14% per annum
be Rs168.The sum due is?
Sol: Time=2yrs,
Rate=14%
Truediscount=Rs168,Amount=x
TrueDiscount=(Amount*R*T)/(100+R*T)
168 =(x*14*2)/(100+(14*2))
=>x=Rs.768
3.The truediscount on Rs.2562 due 4 months hence is Rs.122.The
rate % is?
Sol: Amount=Rs.2562
Time=4/12yrs
TrueDiscount=Rs122
Rate=X%
TrueDiscount=(Amount*R*T)/(100+R*T)
=>122=(2562*x*(4/12))/(100+(x*(4/12))
=>36600+122x=2562x
=>Rate=15%
4.The Truediscount on Rs1760 due after a certain time at
12% per annum is Rs160The time after which it is due is:
Sol: Amount=Rs.1760
Rate=12%
Truediscount=Rs160
Time=x
TrueDiscount=(Amount*R*T)/(100+R*T)
=>x*11*12=100+12x
=>x=5/6yr
Time=10months
5.The interest on Rs.760 for 2yrs is the same as the
Truediscount on Rs960 due 2yrs hence .If the rate of
interest is same in both cases,it is
sol:
Principal=Rs.760
Time=2yrs
amount=Rs960
P*T*R/100=(Amount*R*T)/(100+R*T)
=>(750*2*r)/100=(960*r*2)/(100+2r)
=>150r=2100
rate=14%
6.The SimpleInterest & TrueDiscount on a certain sum of
money for a given time & at a given rate are Rs85
& Rs.80.The sum is.............
sol:
S.I= Rs.85
TrueDiscount=Rs80
Sum=(S.I.*TD)/(SI-T.D)
=Rs.1360
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7.A trader owes a maerchant Rs10028 due 1yr hence The trader
wants to settle the account after 3 months .If the rate of
interest is 12%per annum How much cash should he pay?
Sol:
Time=1yr but he settles account after 3months
so time = 9months
Cash to pay=(Amount*R*T)/(100+R*T)
=(10028*100)/(100+12*(9/12))
=Rs.9200
8.A man buys a watch for Rs1950 in cash and sells it
for Rs2200 at a credit of 1yrThe rate of interest is 10%
per annum Then he gains / loose---------------amount?
Sol:
If he sells it for Rs2200 at a credit of 1yr then the
present worth of that amount is
Presentworth = (Amount*100) /(100+(R*T))
=(2200*100)/(100+10*1)
=2000/-
So he gains 2000-1950=Rs.50
9.A owes Rs.1573 payable 11/2yrs hence ,Aso B owes A Rs.144450
payabe 6months hence If they want to settle the accunt forth
with ,keeping 14% as the rate of interest ,then who should pay
& how much?
Sol:
B->A-----------Amount=Rs.1573
Rate=14%
Time=3/2
Presentworth = (Amount*100) /(100+(R*T))
=(1573*100)/(100+(3/2)*14)
=Rs1300
A->B------------Amount=Rs.1444.50
Rate=14%
Time=1/2
Presentworth = (Amount*100) /(100+(R*T))
=(1444.50*100)/(100+(1/2)*14)
=Rs1350
so B should pay Rs.1350
10.If Rs20 is the TrueDiscount on Rs260due after a certain time.
What wil be trueDiscount on same sum due after ?? of former time,
the rate of interest beng the same?
Sol:
Simple Interest on (260-20) for a gven time=Rs20
Simple Interest on (260-20) for a half time=Rs20*(1/2)
=Rs10
True Discount on Rs.250=Rs10
True Discount on Rs.260=Rs10*260/250
=Rs10.40
11.A has to pay Rs.220 to B after 1yr.B asks A to pay Rs.110
in cash and defer the payment of Rs,.110 for 2 yrs.A agrees
to it.If the rate of interest be 10% per annum in this mode
of payment
sol:
A has to pay =Present worth of Rs220 due 1yr hence
=Rs.(220*100)/(100+(10*1))
=Rs.200 A actually pays=Rs.110+Presentworth of Rs110 due 2 yrs hence
=[110+ (Amount*100) /(100+(R*T))]
=[110+((110*100)/(100+(10*2)))]
=Rs192.66
So A gains Rs(200-192.66)=Rs.7.34
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<b>Complex Problems</b>
1.If Rs.10 be allowed as truedscount on a bill of Rs.110 at the
end of a certaimn time ,then discount allowed on the same sum
due at the end of double the time..
sol:
Amount=Rs110
TrueDiscount=Rs10
Present worth=Amount-TrueDiscount
=Rs110-10
=Rs.100
SI on Rs.100 for a certain time =Rs.10
SI on Rs.100 for doube the time =Rs.20
TrueDiscount on (100+20)=120-100
=Rs20
TrueDiscount on Rs.110 =(110*20)/120
=Rs18.33
2.A man wants to se hs scooter .There are two offers one at
Rs12000 cash and other at acredit of Rs12880 to be paid
after 8 months ,money being at 18% per annum which is better offer?
Sol:
offer1=Rs12000
offer2 Present worth= (Amount*100) /(100+(R*T))
=Rs.12880*100/(100+(18*(8/12))
=Rs11,500
The first offer is better as if he gains 500/-
if he sells at Rs.12000 is better
3.Goods were bought for Rs.600 and sold the same day for
Rs.688.80 at a credit of 9 months and thus gaining 2%
rate of interest per annum is
sol:
MethodI:
Amount=Rs.688.50
Time=9.12
Gaining 2%----------->Presentworth=102%of 600=(102*600)/100
=Rs.612
Presentworth=(Amount*100)/(100+(T*R))
612=(688.80*100)/(100+((9/2)*R))
=>R=162/3%
MethodII:
TrueDiscount=688.50-612
=Rs.76.50
Rate=TrueDiscount*100/(P.W*T)
=(76.50*100)/(612*(9/12))
=162/3%
4.The present worth of Rs.1404 due in 2 equal halfyearly instalments
at 8% per annu8m S.I is
SOl:
Presentworth=(Amount*100)/(100+(T*R))
PresentWorth=Presentworth of Rs.702 6 months hence +
Presentworth of Rs.702 1yr hence
= (702*100)/(100+(1/2*8)) + (702t*100)/(100+(1*8))
=Rs.1325
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