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# In a large housing complex, there are $15$ small buildings and $22$ a large building. Each of the large buildings has $10$ floors with $2$ apartments on each floor. Each of the small buildings has $12$ floors with $3$ apartments on each floor. How many apartments are there in all?

Last updated date: 02nd Aug 2024
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Hint: We will solve this question by calculating the total number of apartments in all buildings and after that calculating the total number of apartments which equals the addition of the total number of apartments in small buildings and the total number of apartments in large buildings.

Complete step-by-step solution:
Step 1: As per the given information in the question, the total number of large buildings are $22$ in which each building has $10$ floors with $2$ apartments on each floor.
So, we will calculate the total number of floors in all $22$ large buildings as shown below:
${\text{Total number of floors}}\left( {{\text{large}}} \right) = 22 \times 10$
By multiplying on the RHS side of the above expression we get:
$\Rightarrow {\text{Total number of floors}}\left( {{\text{large}}} \right) = 220$
Now as we know that each floor has $2$ apartments so we will calculate the total number of apartments as shown below:
$\Rightarrow {\text{Total number of apartments}}\left( {{\text{large}}} \right) = 220 \times 2$
By multiplying on the RHS side of the above expression we get:
$\Rightarrow {\text{Total number of apartments}}\left( {{\text{large}}} \right) = 440$ ………………….. (1)
Step 2: Now, we know that the total number of small buildings are $15$ in which each building has $12$ floors with $3$ apartments on each floor.
So we will calculate the total number of floors in all $15$ small buildings as shown below:
${\text{Total number of floors}}\left( {{\text{small}}} \right) = 15 \times 12$
By multiplying on the RHS side of the above expression we get:
$\Rightarrow {\text{Total number of floors}}\left( {{\text{small}}} \right) = 180$
Now as we know that each floor has $3$ apartments so we will calculate the total number of apartments as shown below:
$\Rightarrow {\text{Total number of apartments}}\left( {{\text{small}}} \right) = 180 \times 3$
By multiplying on the RHS side of the above expression we get:
$\Rightarrow {\text{Total number of apartments}}\left( {{\text{small}}} \right) = 540$ ………………….. (2)
Step 3: Now, calculating the total number of apartments by adding the total number of apartments in large and small buildings as shown below:
After adding equation (1) and (2) we get:
$\Rightarrow {\text{Total number of apartments}} = 440 + 540$
After doing addition in the RHS side of the above expression we get:
$\Rightarrow {\text{Total number of apartments}} = 980$

Total number of apartments are $980$.

Note: Students need to take care while calculating the total number of apartments in each case, as we need to do multiplication instead of addition. For example, if $n$ buildings have $x$ floors then the total number of floors will be equal to $nx$.