Question

The second of two numbers is 6 times the first. The sum is 77. How do you find the numbers?

Answer 1

Hint: To solve these types of questions, we need to form equations according to the given questions. The number of equations and the number of different variables we need to use equals the number of conditions, we are given.

Complete step by step answer:
Here we are given two conditions, the first condition states that, out of two numbers the second one is 6 times the first. And the second condition states that the sum of the given two numbers is 77.
As we are given two conditions, we need to take two variables to make the equations. Let the two numbers are \[x\And y\] respectively. The second one of these is \[y\], according to the first condition, we can say that
\[\Rightarrow y=6x\]
Now, according to the second condition, we can say that
\[\Rightarrow x+y=77\]
We formed two-equation according to the given conditions. We can find the two numbers by solving these two equations. The first equation we formed is \[y=6x\], and the second is \[x+y=77\].
According to the first equation, we have \[y=6x\]. Substituting this relation in the second relation, we get
\[\Rightarrow x+y=77\]
\[\Rightarrow x+6x=77\]
\[\Rightarrow 7x=77\]
Dividing the above equation by 7, we get
\[\begin{align}
  & \Rightarrow \dfrac{7x}{7}=\dfrac{77}{7} \\
 & \therefore x=11 \\
\end{align}\]
Using the first equation, we know \[y=6x\]. Substituting the value of \[x\], we get \[y=6(11)=66\].

Hence, the two numbers are \[11\And 66\] respectively.

Note: We can check if the answer is correct or not by verifying the conditions. According to the first condition, the second number is 6 times the first. Here the second number is \[66\], and the first is \[11\], \[66=6\times 11\] so the first condition is satisfied. The second condition states the sum is \[77\], as \[11+66=77\] the second condition is also satisfied.