5. Percentages, Ratios, Simple & Compound Interest
Percentages
With percentages, you answer the question, "how much out of 100?"
e.g. \$3/5\$ is how much percent? We know \$3/5=(3*20)/(5*20)=60/100\$ or \$60%\$ or \$60\$ percent.
e.g. what is \$20%\$ of \$20\$? \$20%\$ is \$20/100\$, so \$20/100*20=20/5=4\$.

Ratios
If two things increase or decrease proportionately, then their ratio can be used to answer questions about the relationship between those quantities.
Let us illustrate this by means of an example.
e.g. John finishes a task in 4 days. Ann does the same in 6. How long will they take to finish it together?
John finishes \$1/4\$ of his task each day. Ann finishes \$1/6\$ in one day. Together they finish \$1/4+1/6={1*6+1*4}/{6*4}=10/24\$ of a task in one day.
So it takes them \$24/10\$ or \$2.4\$ days to finish the task together.
It is often the case that we can reason effectively to solve problems involving ratios by first determining whether the answer should be larger or smaller than individual quantities.
Here is it pretty clear that if John and Ann work together, they finish faster than either one alone. That should be a sanity check we use.
e.g. Water flows into a tank at 30 gallons a minute. The tank leaks 30 gallons an hour. What is the net inflow of water into the tank in an hour?
Well, 30 gallons flows in per minute, or \$30*60=1800\$ gallons in the hour, from which only 30 gallons flow out, so 1770 gallons is the net inflow.

Simple Interest
Interest is what you are paid when you keep a deposit on a bank. It helps your money grow. Typically it is paid annually.
e.g. you are paid 10% interest on your savings account of \\$1000 for 3 years. This gives you \$1000*10/100=\\$100\$ interest per year
Over three years, you receive \$100*3=\\$300\$ in interest on your principal of \\$1000.
The total amount you have in the bank is \$\\$1300 (=\\$1000+\\$300)\$

Compound interest compounds the calculation by including each year's interest as it is paid out, in the future years' interest calculation.
Thus, your interest as well as amount rise quite quickly with compounding.
e.g. for the same bank account as earlier, with compounding we get:
at the end of year 1: principal=\\$1000, interest=\$10%*1000=\\$100\$, amount=\\$1100
at the end of year 2: principal=\\$1100, interest=\$10%*1100=\\$110\$, amount=\\$1210
at the end of year 3: principal=\\$1210, interest=\$10%*1210=\\$121\$, amount=\\$1320
So net interest paid in 3 years with (annual) compounding=\$\\$1320-\\$1000=\\$320\$
This, as you can see, is \\$20 more than the \\$300 we got with simple interest.

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