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Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to 21 times the largest of the seven integers. What is the smallest integer that Pearl wrote down?

 May 6, 2023
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Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to 21 times the largest of the seven integers. What is the smallest integer that Pearl wrote down?   

 

Call the smallest integer x.  Then each following integer  

will be 1 larger than the integer that preceded it.  

 

x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) + (x + 6)  =  21 • (x + 6)  

 

                                                                          7x + 21  =  21x + 126  

 

Combine like terms                                                –14x  =  105   

 

Divide both sides by –14                                             x  =  –7.5  

 

The problem fails already, since 7.5 is not  

an integer, whether it's positive or negative.            NO SOLUTION  

 

But let's press ahead and see if this would have worked.   

Check answer by substituting the value into the problem.   

 

(–7.5) + (–7.5 + 1) + (–7.5 + 2) + (–7.5 + 3) + (–7.5 + 4) + (–7.5 + 5) + (–7.5 + 6)  =  (21)(–7.5 + 6) 

 

(–7.5) + (–6.5) + (–5.5) + (–4.5) + (–3.5) + (–2.5) + (–1.5)  =  (21)(1.5) 

 

                                                                                  –31.5  =  –31.5  

 

So, –7.5 would have been a good answer IF the problem

had not specified that the numbers have to be integers.  

.

 Jun 30, 2023

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