Answer:
Fraction of the people chose a tart =  
 
Step-by-step explanation:
Given - A group of people went to a coffee shop. Each person chose a 
             drink  and a treat. number of people who chose coffee: number of  
             people who chose tea = 1:4. Of those who chose coffee,  number of  
             people who chose a muffin number of people who chose a tart =    
             2:7. Of  those who chose tea,  number of people who chose a muffin  
             : number of people who chose a tart = 5:1.
To find - What fraction of the people chose a tart?
Proof -
Let the Total number of People went to a coffee shop = 5x
Now,
As given, number of people who chose coffee: number of people who chose tea = 1:4
⇒Total number of people who chooses coffee =  = x
 = x
   Total number of people who chooses tea  =  = 4x
 = 4x
Now,
As given, 
Of those who choose coffee,  number of people who chose a muffin number of people who chose a tart = 2:7
⇒Total number of people who choose muffin after coffee = 
    Total number of people who choose tart after coffee = 
Also,
Given that 
Of those who chose tea,  number of people who chose a muffin : number of people who chose a tart = 5:1
⇒Total number of people who choose muffin after tea = 
    Total number of people who choose tart after tea = 
∴ we get
Total number of people who chooses tart = Total number of people who choose tart after coffee + Total number of people who choose tart after tea
⇒Total number of people who chooses tart =  +
 +  
 
                                                                          = 
                                                                          = 
                                                                          = 
⇒Total number of people who chooses tart = 
Now,
Fraction of the people chose a tart = 
                                                            = 
                                                            =  
 
⇒Fraction of the people chose a tart = 