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Fudgin [204]
3 years ago
7

Two certificates of deposit pay interest that differ by 3%. Money invested for one year in the first CD earns $240 interest. The

same principal invested in the other CD earns $360. Find the two rates of interest. Smaller value and Larger value (%)
Mathematics
1 answer:
Soloha48 [4]3 years ago
3 0

a = interest rate of first CD

b = interest rate of second CD

and again, let's say the principal invested in each is $X.

\bf a-b=3\qquad \implies \qquad \boxed{b}=3+a~\hfill \begin{cases} \left( \frac{a}{100} \right)X=240\\\\ \left( \frac{b}{100} \right)X=360 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \left( \cfrac{a}{100} \right)X=240\implies X=\cfrac{240}{~~\frac{a}{100}~~}\implies X=\cfrac{24000}{a} \\\\\\ \left( \cfrac{b}{100} \right)X=360\implies X=\cfrac{360}{~~\frac{b}{100}~~}\implies X=\cfrac{36000}{b} \\\\[-0.35em] ~\dotfill\\\\

\bf X=X\qquad thus\qquad \implies \cfrac{24000}{a}=\cfrac{36000}{b}\implies \cfrac{24000}{a}=\cfrac{36000}{\boxed{3+a}} \\\\\\ (3+a)24000=36000a\implies \cfrac{3+a}{a}=\cfrac{36000}{24000}\implies \cfrac{3-a}{a}=\cfrac{3}{2} \\\\\\ 6-2a=3a\implies 6=5a\implies \cfrac{6}{5}=a\implies 1\frac{1}{5}=a\implies \blacktriangleright 1.2 = x\blacktriangleleft

\bf \stackrel{\textit{since we know that}}{b=3+a}\implies b=3+\cfrac{6}{5}\implies b=\cfrac{21}{5}\implies b=4\frac{1}{5}\implies \blacktriangleright b=4.2 \blacktriangleleft

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For the next unique toy, there is now a 3 in 5 chance of getting it:

E_3 = (\frac{3}{5})^{-1} = 1.67

Following that logic, there is a 2 in 5 chance of getting the 4th unique toy:

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Finally, there is a 1 in 5 chance to get the last unique toy:

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The expected number of boxes to obtain a full set is:

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