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At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to $3.15. Gary orders a bagel and a muffin, which comes out to $3.50. Larry orders a piece of toast, two bagels, and three muffins, which comes out to $8.15. How many cents does one bagel cost?

 Jun 1, 2024
 #1
avatar+850 
+2

 

At a cafeteria,   

Mary orders two pieces of toast and a bagel, which comes out to $3.15.   

Gary orders a bagel and a muffin, which comes out to $3.50.   

Larry orders a piece of toast, two bagels, and three muffins, which comes out to $8.15.   

How many cents does one bagel cost?   

 

                                           Mary          2T + B = 315 ............. —>  T = (315 – B) / 2       

                                           Gary          B + M = 350 ................—>  M = (350 – B)      

                                           Larry          T + 2B + 3M = 815  

 

                                           Substitute for Toast & Muffin into Larry's order  

 

                                           [(315 – B) / 2]  +  2B  +  [3 (350 – B)]  =  815   

 

                                           [(315 – B) / 2]  +  2B  +  (1050 – 3B)  =  815   

 

Multiply both sides by 2       (315 – B)  +  4B  +  (2100 – 6B)  =  1630    

 

Combine like terms              –3B + 2415  =  1630     

 

                                                        –3B  =  – 785     

 

                                                            B  =  261 2/3  ¢    

 

They usually construct these problems to come out with an even number.   

I've gone back over the steps a hundred times and cannot find a mistake.    

.

 Jun 2, 2024
 #2
avatar+806 
+1

From the problem, we can write the system

\(2T + 1B = 3.15 \\ 1M + 1B = 3.50 \\ 1T + 2B + 3M = 8.15 \)

 where t is toast, b is bagel and m is muffin. 

 

From the first two equations, we have

\( [3.15 - B ] / 2 = T \\ M = 3.50 - B\)

 

Subbing this in to the third equation, we have 

\([ 3.15 - B ] / 2 + 2B + 3 [ 3.50 - B ] = 8.15 \\ [3.15 - B ] + 4B + 6 [ 3.50 -B ] = 16.30 \\ -3B + 3.15 + 21 = 16.30 \\ -3B = 16.30 - 3.15 - 21 \\ -3B = -7.85 \\ B = 7.85 / 3 ≈ $2.62\)

 

This is about 262 cents. 

 

Thanks! :)

 Jun 2, 2024

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