Estimating quantities

Estimating quantities

Remember:

• Positive exponents: the number gets bigger as the exponent gets bigger

• Negative exponents: The number gets smaller as the exponent gets bigger/more negative

What is a ratio?

A ratio compares two numbers

• The ratio a to b is written:

   $\frac{a}{b}$  

• We can calculate the ratio by dividing the fraction

Open Ended

Remember: When we divide two numbers in exponential form

 $\frac{10^a}{10^b}$  how can we combine the exponents?

New York City

1723

Population: 7,248

New York City

1870

Population: 942,292

Comparing the populations

• Population in 1723: 7,248

• Estimate: is this number closer to

   $10^3$  or  $10^4$  ?

• Population in 1870: 942,292

• Estimate: is this number closer to  $10^5$  or  $10^6$  ?

Multiple Choice

Calculate the ratio:

 $\frac{10^6}{10^4}$  

•  $10^2$  

•  $10^1$  

•  $10^{10}$  

Comparing the populations gets us

 $10^2$  as the solution to our ratio:

That means NYC's population in 1870 was  $10^2$  times bigger than in 1823
(In this time, the population grew by 100 times)

If we hadn't estimated...

We would have gotten 130 as our answer instead of 100

How can we be more accurate?

• To estimate 7,248 we could say 7,000 instead of 10,000

• To write this with exponents: $7000\ =\ 7\times10^3$  

  
  

Practice:

• Population of NYC now: 8,336,697. Round to the nearest million

• 8,000,000

•  $8\times1,000,000\ =\ 8\times10^6$  

Practice:

• Population of New York State now: 19,570,261. Round to the nearest million

• 20,000,000

•  $20\times1,000,000\ =\ 2\times10,000,000=\ 2\times10^7$  

Comparing New York State to NYC

•  $\frac{state\ population}{city\ population}=\frac{2\times10^7}{8\times10^6}=\frac{2}{8}\times\frac{10^7}{10^6}$  

• What is  $\frac{2}{8}$  ?

• What is  $\frac{10^7}{10^6}$  ?

• Our answer:  $4\times10^1=4\times10=40$  

• New York State's population is 40 times the population of NYC