Question

A town's population increased by $1200$ people, and then this new population decreased 11%. The town now had $32$ less people than it did before the $1200$ increase. Find the original population.
(A). $10000$
(B). $12000$
(C). $15000$
(D). $25000$

Answer 1

Hint: We want to find the original population. Assume the original population to be $x$. After that, stepwise apply the conditions mentioned in the problem. Simplify it and you will get the answer.

Complete step-by-step answer:

In Mathematics, a percentage (from Latin per centum "by a hundred") is a number or ratio expressed as a fraction of $100$. It is often denoted using the percent sign, "%", or the abbreviations "pct.", "pct"; sometimes the abbreviation "pc" is also used. A percentage is a dimensionless number (pure number).
Due to inconsistent usage, it is not always clear from the context what a percentage is relative to. When speaking of a "10% rise" or a "10% fall" in a quantity, the usual interpretation is that this is relative to the initial value of that quantity. For example, if an item is initially priced at $200$ and the price rises 10% (an increase of 20%), the new price will be $220$. Note that this final price is 110% of the initial price (100%+10%= 110%).
Some other examples of percent changes:
An increase of 100% in a quantity means that the final amount is 200% of the initial amount (100% of initial + 100% of increase = 200% of initial); in other words, the quantity has doubled.
An increase of 800% means the final amount is 9 times the original (100%+ 800%= 900%=9 times as large).
A decrease of 60% means the final amount is 40% of the original (100% – 60% = 40%).
A decrease of 100% means the final amount is zero (100% – 100% = 0%).
In general, a change of x percent in a quantity results in a final amount that is 100 + x percent of the original amount (equivalently, (1 + 0.01x) times the original amount).
Let us assume the original population $=x$.
So when the population is increased by 1200 people $=x+1200$.
After that, the population decreased by 11% = population increased by 1200 people – 11% of population increased by the people.
Population decreased 11%
$=(x+1200)-\left( \dfrac{11}{100} \right)(x+1200)$
The town now had 32 less people than it did before the 1200 increase
$=(x+1200)-\left( \dfrac{11}{100} \right)(x+1200)=x-32$.
Now simplifying above we get,
$\begin{aligned} & (x+1200)\left( 1-\left( \dfrac{11}{100} \right) \right)=x-32 \\\ & (x+1200)\left( \dfrac{89}{100} \right)=x-32 \\\ & (89x+106800)=100x-3200 \\\ & 11x=110000 \\\ \end{aligned}$
$x=10000$
So we get the original population as 10000.

Note: Read the question carefully. Also, take care that confusion does not occur. While simplifying, do not make silly mistakes. Your concept regarding percentage should be clear. Also, while solving, solve the problem systematically so jumbling does not occur.
Students often make mistakes when subtracting the 11% decrease, i.e., they often subtract 11% of the original population instead of the new population.