Simple Interest

[Mavisa97]

The two capitals, of € 2500 and € 1250 respectively, are used at simple interest, the first at rate i1 for 3 years and the second at rate i2 for two years, and they produce the total interest of € 900. Switching those two rates, the interest rises to € 1100.
Determine the two rates i₁ and i₂?.

Solutions are i₁ = 8% and i₂ = 12%

Here's how I tried to do the math:

[MATH] C1 = 2500 \quad t=3 \\ C2 = 1250 \quad t = 2 \\ I=I1+I2 =900 [/MATH]

Switching the two rates [MATH]i1, i2[/MATH] [MATH] [/MATH][MATH]\quad I = 1100[/MATH]
[MATH] I1 = 2500 * 3 * i1 = 7500i \\\\ I2 = 1250 * 2 * i2 = 2500i \\\\ I = 7500i + 2500i\\\\ 900 = 7500i + 2500i\\\\\ i = \frac{900}{10000} = 0.09 = 9\% [/MATH]
How should I continue to determine the two rates i₁ and i₂?

 

[nasi112]

you have
[MATH]7500i_1 + 2500i_2 = 900[/MATH]and
[MATH]7500i_2 + 2500i_1 = 1100[/MATH]
Solving the two equations will give you [MATH]i_1[/MATH] and [MATH]i_2[/MATH]

 

[Mavisa97]

Hi and thanks for the quick answer.
I tried to solve the first equation but the results are negative interests numbers...
Should you show me the correct method to solve it?

Solving for [MATH]i_1[/MATH]
[MATH] 7500i_1+2500i_2=900 [/MATH]
[MATH]i_1= \frac{900-2500}{7500} = -0.2133333i_2 [/MATH]
Solving for [MATH]i_2[/MATH]
[MATH] 7500i_2+2500i_1=900 [/MATH]
[MATH]i_2= \frac{900-7500}{2500} = -2.64i_1 [/MATH]

 

[Deleted member 4993]

Hi and thanks for the quick answer.
I tried to solve the first equation but the results are negative interests numbers...
Should you show me the correct method to solve it?

Solving for [MATH]i_1[/MATH]
[MATH] 7500i_1+2500i_2=900 [/MATH]
[MATH]i_1= \frac{900-2500}{7500} = [S]-0.2133333i_2[/S] [/MATH] ..........................................INCORRECT

Solving for [MATH]i_2[/MATH]
[MATH] 7500i_2+2500i_1=900 [/MATH]
[MATH]i_2= \frac{900-7500}{2500} = -2.64i_1 [/MATH]

Correct step:

[MATH] 7500i_1+2500i_2=900 [/MATH]
[MATH]i_1= \frac{900-2500*i_2}{7500} [/MATH] .................... (3)

You CANNOT isolate i_2 at this step.

Now replace i_1 in your second equation - and you can isolate i_2 there.

continuing:

\(\displaystyle 7500*i_2+2500*i_1=1100\)

\(\displaystyle 7500*i_2 + 2500* \frac{900 - 2500*i_2}{7500} \ = \ 1100\)

simplify above and isolate i₂ and solve for i₂ - then solve for i₁ from eqn(3)

 

[nasi112]

Correct step:

[MATH] 7500i_1+2500i_2=900 [/MATH]
[MATH]i_1= \frac{900-2500*i_2}{7500} [/MATH] .................... (3)

You CANNOT isolate i_2 at this step.

Now replace i_1 in your second equation - and you can isolate i_2 there.

continuing:

\(\displaystyle 7500*i_2+2500*i_1=1100\)

\(\displaystyle 7500*i_2 + 2500* \frac{900 - 2500*i_2}{7500} \ = \ 1100\)

simplify above and isolate i₂ and solve for i₂ - then solve for i₁ from eqn(3)

I was about to write what Khan has written

Thank you Mr. Khan

 

[Mavisa97]

Thank you very much.

[MATH]7500i_2+2500i_1=1100[/MATH]
[MATH]7500i_2+2500*\frac{900-2500i_2}{7500}=1100[/MATH]
[MATH]7500i_2+300-833.333i_2=1100[/MATH]
[MATH]7500i_2-833.333i_2=1100-300[/MATH]
[MATH]6666.667i_2=800[/MATH]
[MATH]i_2=\frac{800}{6666.667}=0.119 =12\%[/MATH]

[MATH]i_1=\frac{900-2500*0.119}{7500}=0.08=8\%[/MATH]

 

[nasi112]

Bravo, you did a great job Mavisa?