Question

Question:

A man has $ 10,000 to invest. He invests $ 4000 at 5 % and $ 3500 at 4 %. In order to have a yearly income of $ 500, he must invest the remainder at:

(a) 6 % , (b) 6.1 %, (c) 6.2 %, (d) 6.3 %, (e) 6.4 %​

Answer 1

★ Concept ::

Here the concept of Simple Interest has been used. We see that we are given the total money that is to be invested by a man. Even we are given different amount out of total money which are invested at different rates. Firstly we can find the income from the different smaller sums of money. Then we can subtract it from total income. Then we have to find the remainder money and then using the formula of Simple Interest, we can find the remaining rate.

Let's do it !!

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★ Formula Used :-

$\\\;\boxed{\sf{\red{Income\;=\;\bf{Rate\;\times\;Money}}}}$

$\\\;\boxed{\sf{\pink{S.I.\;=\;\bf{\dfrac{P\:\times\:R\:\times\:T}{100}}}}}$

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★ Solution :-

Given,

» Total Money = $ 10, 000

» Sum of money at first investment = Rs. 4000

» Sum of money at second investment = Rs. 3500

» Rate of first investment = 5 %

» Rate of second investment = 4 %

» The remainder money = Total money - (all the investments) = 10000 - (4000 + 3500) = 1000 - 7500 = 2500

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~ For the income in both investments ::

By formula we know that,

$\\\;\sf{\rightarrow\;\;Income\;=\;\bf{Rate\;\times\;Money}}$

• During first investment ::

By applying values in the above formula, we get

$\\\;\sf{\rightarrow\;\;Income\;in\;I^{st}\;Investment\;=\;\bf{5\%\;\times\;4000}}$

$\\\;\sf{\rightarrow\;\;Income\;in\;I^{st}\;Investment\;=\;\bf{\dfrac{5}{100}\;\times\;4000}}$

$\\\;\sf{\rightarrow\;\;Income\;in\;I^{st}\;Investment\;=\;\bf{5\;\times\;40}}$

$\\\;\bf{\rightarrow\;\;Income\;in\;I^{st}\;Investment\;=\;\bf{\blue{200\;\;\ }}}$

• During second investment ::

By applying values, in the formula we get,

$\\\;\sf{\rightarrow\;\;Income\;in\;II^{st}\;Investment\;=\;\bf{4\%\;\times\;3500}}$

$\\\;\sf{\rightarrow\;\;Income\;in\;II^{st}\;Investment\;=\;\bf{\dfrac{4}{100}\;\times\;3500}}$

$\\\;\sf{\rightarrow\;\;Income\;in\;II^{st}\;Investment\;=\;\bf{4\;\times\;35}}$

$\\\;\bf{\rightarrow\;\;Income\;in\;II^{st}\;Investment\;=\;\bf{\green{140\;\;\ }}}$

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~ For income to be earned more ::

We see that we got incomes in the first two investments. If we subtract their sum from the total yearly income then we can get the left remainder income. This remainder income can be used to calculate the rate of final investment of remainder.

✒ Remainder Income = Total Income (yearly) - Income already earned (first two investments)

✒ Remainder Income = 500 - (200 + 140)

✒ Remainder Income = 500 - 340

✒ Remainder Income = $ 160

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~ For the rate of the investment of the remainder ::

We have got,

• Principal = P = The Remainder sum = $ 2500

• Rate = rate of investment = R %

• Time = for yearly income = T = 1 year

• S. I. = income to be earned = $ 160

By using the formula of Simple Interest, we get

$\\\;\sf{\Longrightarrow\;\;S.I.\;=\;\bf{\dfrac{P\:\times\:R\:\times\:T}{100}}}$

By applying values, we get

$\\\;\sf{\Longrightarrow\;\;160\;=\;\bf{\dfrac{2500\:\times\:R\:\times\:1}{100}}}$

$\\\;\sf{\Longrightarrow\;\;160\;=\;\bf{25\:\times\:R\:\times\:1}}$

$\\\;\sf{\Longrightarrow\;\;160\;=\;\bf{25\:\times\:R}}$

$\\\;\sf{\Longrightarrow\;\;R\;=\;\bf{\dfrac{160}{25}}}$

$\\\;\sf{\Longrightarrow\;\;R\;=\;\bf{6.4\;\;\%}}$

Thus, this is the answer. Let's see the options to see correct option.

So, the correct option is option e.) 6.4 %

$\\\;\underline{\boxed{\tt{Required\;\:rate\;\:of\;\:investment\;=\;\bf{\purple{6.4\;\:\%}}}}}$

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★ More to know :-

$\\\;\sf{\leadsto\;\;Amount\;=\;Principal\;+\;Interest}$

$\\\;\sf{\leadsto\;\;C.I.\;=\;P\bigg(1\;+\;\dfrac{R}{100}\bigg)^{T}\;-\;P}$

Answer 2

Answer:

ꜱᴏʟᴜᴛɪᴏɴ:

ɪɴᴄᴏᴍᴇ ꜰʀᴏᴍ $ 4000 ᴀᴛ 5 % ɪɴ ᴏɴᴇ ʏᴇᴀʀ = $ 4000 ᴏꜰ 5 %.

= $ 4000 × 5/100.

= $ 4000 × 0.05.

= $ 200.

ɪɴᴄᴏᴍᴇ ꜰʀᴏᴍ $ 3500 ᴀᴛ 4 % ɪɴ ᴏɴᴇ ʏᴇᴀʀ = $ 3500 ᴏꜰ 4 %.

= $ 3500 × 4/100.

= $ 3500 × 0.04.

= $ 140.

ᴛᴏᴛᴀʟ ɪɴᴄᴏᴍᴇ ꜰʀᴏᴍ 4000 ᴀᴛ 5 % ᴀɴᴅ 3500 ᴀᴛ 4 % = $ 200 + $ 140 = $ 340.

ʀᴇᴍᴀɪɴɪɴɢ ɪɴᴄᴏᴍᴇ ᴀᴍᴏᴜɴᴛ ɪɴ ᴏʀᴅᴇʀ ᴛᴏ ʜᴀᴠᴇ ᴀ ʏᴇᴀʀʟʏ ɪɴᴄᴏᴍᴇ ᴏꜰ $ 500 = $ 500 - $ 340.

= $ 160.

ᴛᴏᴛᴀʟ ɪɴᴠᴇꜱᴛᴇᴅ ᴀᴍᴏᴜɴᴛ = $ 4000 + $ 3500 = $7500.

ʀᴇᴍᴀɪɴɪɴɢ ɪɴᴠᴇꜱᴛ ᴀᴍᴏᴜɴᴛ = $ 10000 - $ 7500 = $ 2500.

ᴡᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ, ɪɴᴛᴇʀᴇꜱᴛ = ᴘʀɪɴᴄɪᴘᴀʟ × ʀᴀᴛᴇ × ᴛɪᴍᴇ

ɪɴᴛᴇʀᴇꜱᴛ = $ 160,

ᴘʀɪɴᴄɪᴘᴀʟ = $ 2500,

ʀᴀᴛᴇ = ʀ [ᴡᴇ ɴᴇᴇᴅ ᴛᴏ ꜰɪɴᴅ ᴛʜᴇ ᴠᴀʟᴜᴇ ᴏꜰ ʀ],

ᴛɪᴍᴇ = 1 ʏᴇᴀʀ.

160 = 2500 × ʀ × 1.

160 = 2500ʀ

160/2500 = 2500ʀ/2500 [ᴅɪᴠɪᴅᴇ ʙᴏᴛʜ ꜱɪᴅᴇꜱ ʙʏ 2500]

0.064 = ʀ

ʀ = 0.064

ᴄʜᴀɴɢᴇ ɪᴛ ᴛᴏ ᴀ ᴘᴇʀᴄᴇɴᴛ ʙʏ ᴍᴏᴠɪɴɢ ᴛʜᴇ ᴅᴇᴄɪᴍᴀʟ ᴛᴏ ᴛʜᴇ ʀɪɢʜᴛ ᴛᴡᴏ ᴘʟᴀᴄᴇꜱ ʀ = 6.4 %

ᴛʜᴇʀᴇꜰᴏʀᴇ, ʜᴇ ɪɴᴠᴇꜱᴛᴇᴅ ᴛʜᴇ ʀᴇᴍᴀɪɴɪɴɢ ᴀᴍᴏᴜɴᴛ $ 2500 ᴀᴛ 6.4 % ɪɴ ᴏʀᴅᴇʀ ᴛᴏ ɢᴇᴛ $ 500 ɪɴᴄᴏᴍᴇ ᴇᴠᴇʀʏ ʏᴇᴀʀ.

ᴀɴꜱᴡᴇʀ: (ᴇ)