Question

What must be added to \[6,10,14\] and \[22\] so that they are in proportion?

Answer 1

Hint: We need to find a number such that when it is added to the given four numbers, the numbers will then be in proportion. When four numbers are in Proportion, then the product of extremes is equal to the product of means. i.e. if \[a,b,c,d\] are in proportion, then \[a \times d = b \times c\]. So now, we will suppose that the number to be added is \[x\]. Then, we will add this number \[x\] to each of the four numbers given and then, we are given that when this number is added to the given numbers, the numbers will be in proportion. After adding \[x\], we will then use the property of proportion i.e. product of extremes is equal to product of means to find the value of \[x\].

Complete step-by-step solution:
Let the number to be added is equal to \[x\].
Now, let us add this number \[x\] to all the given numbers. i.e. \[6,10,14,22\]
Therefore, after adding \[x\] to the numbers, the numbers become \[6 + x,10 + x,14 + x,22 + x\]. We are given that when this number \[x\] is added to \[6,10,14,22\], then these numbers will be in proportion. Hence, we can say that \[6 + x,10 + x,14 + x,22 + x\] are in proportion.
Now, using the property of the proportional numbers, we have
Product of extremes = Product of means
Product of \[6 + x\] and \[22 + x\]= Product of \[10 + x\] and \[14 + x\]
i.e. \[(6 + x)(22 + x) = (10 + x)(14 + x)\]
\[6(22 + x) + x(22 + x) = 10(14 + x) + x(14 + x)\] (Opening the brackets)
\[\Rightarrow 6 \times 22 + 6x + 22x + {x^2} = 10 \times 14 + 10x + 14x + {x^2}\]
\[\Rightarrow 132 + 6x + 22x + {x^2} = 140 + 10x + 14x + {x^2}\]
\[\Rightarrow 132 + 28x + {x^2} = 140 + 24x + {x^2}\]
\[\Rightarrow 132 + 28x + {x^2} - {x^2} - 24x = 140\] (Shifting the terms)
\[\Rightarrow 28x + {x^2} - {x^2} - 24x = 140 - 132\]
 \[\Rightarrow 4x = 8\]
 \[\Rightarrow x = \dfrac{8}{4} = 2\]
Therefore, \[2\] is to be added to \[6,10,14,22\] so that they are in proportion.

Note: We must be careful while framing the equation so that we don’t have to change the order of the numbers, otherwise the proportion will be disturbed. Also, we need to make sure that the number \[x\] is to be added to all the numbers and not just one or two numbers. While we are shifting the terms and while opening the brackets, calculations and changing of signs are to be taken care of. While maintaining order, we have to see which terms are on the extreme end and which terms are in the middle.