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Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to \(\frac{21}{4}\) times the largest of the seven integers. What is the smallest integer that Pearl wrote down?

 Jul 17, 2023
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Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to 21 / 4 times the largest of the seven integers. What is the smallest integer that Pearl wrote down?   

 

Let the first integer be represented by x.   

Since the integers are consecutive, each following   

integer is one more than the integer before it.  

 

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

 

                                                                                21x + 126  

                                                             7x + 21  =  –––––––––   

                                                                                      4          

Multiply both sides by 4    

                                                            28x + 84  =  21x + 126               

Subtract 21x from both sides   

                                                              7x + 84  =            126               

Subtract 84 from both sides                     

                                                                      7x  =  42    

Divide both sides by 7    

                                                                        x  =  6    

 

check answer    

                               6 + 7 + 8 + 9 + 10 + 11 + 12  =  (21/4)(12)  

                                                                       63  =  63   

.

 Jul 17, 2023

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