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# Percent problem

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Ms. Forsythe gave the same algebra test to her three classes. The first class averaged 80%, the second class averaged 85%, and the third 89%. Together, the first two classes averaged 83%, and the second and third classes together averaged 87%. What was the average for all three classes combined? Express your answer to the nearest hundredth.

Jul 18, 2019

#1
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You can draw a Venn Diagram showing the first class, the second class. and the third class.

Since the first two classes averaged 83%, the first class averaged 80% and the second class averaged 85%, we can calculate the overlap between the classes. The average of students in two classes can be calculated by adding the first class's and the second class's averages together, then subtracting the overlap. So, the overlap of the first two classes is 83*2-80-85=1%.

We can do the same thing for the second and third class. The overlap of the second and third class is 87*2-85-89=0%. So, no students go to the second and third classes. Since we also need the overlap between the third and first class and that information is not provided in the question, there is not enough information to calculate the average of all three classes.

Jul 18, 2019
#2
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Ms. Forsythe gave the same algebra test to her three classes.
The first class averaged 80%, the second class averaged 85%, and the third 89%.
Together, the first two classes averaged 83%, and the second and third classes together averaged 87%.
What was the average for all three classes combined?

$$\text{The students in the first class  = s_1 } \\ \text{The students in the second class  = s_2 } \\ \text{The students in the third class  = s_3 } \\ \text{The sum of the points in the first class  = p_1 } \\ \text{The sum of the points in the second class  = p_2 } \\ \text{The sum of the points in the third class  = p_3 } \\ \text{The maximal points of the test  = p }$$

$$\begin{array}{|lrcl|lrcl||lrcl|} \hline & \dfrac{\frac{p_1}{s_1}} {p} &=& 80\% & & \dfrac{\frac{p_2}{s_2}} {p} &=& 85\% & & \dfrac{\frac{p_3}{s_3}} {p} &=& 89\% \\ (1) & p_1 &=& 80\%ps_1 & (2) & p_2 &=& 85\%ps_2 &(3) & p_3 &=& 89\%ps_3 \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \dfrac{\frac{p_1+p_2}{s_1+s_2}} {p} &=& 83\% \quad | \quad p_1 = 80\%ps_1,\ p_2 = 85\%ps_2 \\ \dfrac{\frac{80\%ps_1+85\%ps_2}{s_1+s_2}} {p} &=& 83\% \\ \dfrac{80\%ps_1+85\%ps_2}{(s_1+s_2)p} &=& 83\% \\ \dfrac{80\%s_1+85\%s_2}{(s_1+s_2)} &=& 83\% \\ 80\%s_1+85\%s_2 &=& 83\% (s_1+s_2) \\ 80\%s_1+85\%s_2 &=& 83\%s_1+83\%s_2 \\ 85\%s_2-83\%s_2 &=& 83\%s_1-80\%s_1 \\ 2\%s_2&=& 3\%s_1 \\ 2s_2&=& 3s_1 \\ \mathbf{s_2} &=& \mathbf{1.5s_1} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \dfrac{\frac{p_2+p_3}{s_2+s_3}} {p} &=& 87\% \quad | \quad p_2 = 85\%ps_2,\ p_3 = 89\%ps_3 \\ \dfrac{\frac{85\%ps_2+89\%ps_3}{s_2+s_3}} {p} &=& 87\% \\ \dfrac{85\%ps_2+89\%ps_3}{(s_2+s_3)p} &=& 87\% \\ \dfrac{85\%s_2+89\%s_3}{(s_2+s_3)} &=& 87\% \\ 85\%s_2+89\%s_3 &=& 87\%(s_2+s_3) \\ 85\%s_2+89\%s_3 &=& 87\%s_2+87\%s_3 \\ 89\%s_3-87\%s_3 &=& 87\%s_2-85\%s_2 \\ 2\%s_3 &=& 2\%s_2 \\ \mathbf{s_3} &=& \mathbf{s_2} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \dfrac{\frac{p_1+p_2+p_3}{s_1+s_2+s_3}} {p} &=& x \quad | \quad p_1 = 80\%ps_1,\ p_2 = 85\%ps_2,\ p_3 = 89\%ps_3 \\ \dfrac{\frac{80\%ps_1+85\%ps_2+89\%ps_3}{s_1+s_2+s_3}} {p} &=& x \\ \dfrac{80\%ps_1+85\%ps_2+89\%ps_3}{(s_1+s_2+s_3)p} &=& x \\ \dfrac{80\%s_1+85\%s_2+89\%s_3}{(s_1+s_2+s_3)} &=& x \\ 80\%s_1+85\%s_2+89\%s_3 &=& x(s_1+s_2+s_3) \quad | \quad s_2 =1.5s_1,\ s_3=s_2=1.5s_1 \\ 80\%s_1+85\%1.5s_1+89\%1.5s_1 &=& x(s_1+1.5s_1+1.5s_1) \\ 80\%s_1+85\%1.5s_1+89\%1.5s_1 &=& x(4s_1) \\ 80\% +85\%1.5 +89\%1.5 &=& 4x \\ 80\% +127.5\% +133.5\% &=& 4x \\ 341\% &=& 4x \quad | \quad : 4 \\ 85.25\% &=& x \\ \mathbf{x} &=& \mathbf{85.25\%} \\ \hline \end{array}$$

The average for all three classes combined is $$\mathbf{85.25\%}$$ Jul 18, 2019
#3
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Pick a number (that is easy to work with) for the number of students in 1st class:    say  80 students

(80 *80%  +  2nd * 85%) = (80+2nd)* 83    solve for 2nd class = 120 students

Now do the same for the second and third

120(.85) + 3rd(.89) = (120+3rd)*87     solve for 3rd = 120 students

Now find the overall average:

(80(.8) + 120(.85) + 120(.89)) / 320 = .8525 = 85.25%             (320 would be the total number of students)

Jul 18, 2019
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ElectricPavlov  Jul 18, 2019
#4
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Pick a number (that is easy to work with) for the number of students in 1st class:    say  80 students

(80 *80%  +  2nd * 85%) = (80+2nd)* 83    solve for 2nd class = 120 students

Now do the same for the second and third

120(.85) + 3rd(.89) = (120+3rd)*87     solve for 3rd = 120 students

Now find the overall average:

(80(.8) + 120(.85) + 120(.89)) / 320 = .8525 = 85.25%             (320 would be the total number of students)

Jul 18, 2019