## Friday, December 14, 2012

### Ratio, Proportions -- Beginning Problem Soving (SAT level)

Questions:
#1: At a county fair, if adults to kids ratio is 2 to 3 and there are 250 people at the fair. How many adults and how many kids?

Solution I :  Adults to Kids = 2 : 3 (given). Let there be 2x adults and 3x kids (only when two numbers have the same multiples can you simplify the two number to relatively prime.
2x + 3x = 250;  5x = 250; x = 50 so there are 2x or 2 * 50 = 100 adults and 3x or 3 * 50 = 150 kids.

Solution II:  From the given information, you know that every time when there are 5 parts (2 + 3 = 5), there
will be 2 parts for A, or $\frac{2}{5}}$  of the total, and 3 parts for B, or $\frac{3}{5}}$ of the total.
$\frac{2}{5}}$ * 250 = 100 adults and $\frac{3}{5}}$ * 250 = 150 kids

#2: In a mixture of peanuts and cashews, the ratio by weight of peanuts to cashews is 5 to 2. How many pounds of cashews will there be in 4 pounds of this mixture? (an actual SAT question)

Solution:
You can use either the first or second method (see above two solutions) but the second solution is much faster.
$\textcolor{red}{\frac{8}{7}}$ pounds

#3: Continue with question #1: How many more kids than adults go to the fair?

Solution I:
Use the method on #1 and then find the difference.
150 - 100 = 50 more kids.

Solution II:
Using the method for #1: solution II, you know  $\frac{2}{5}}$  of the all the people go to the fair are adults ,
and $\frac{3}{5}}$ of the total people go to the fair are kids.
$\frac{3}{5}}$ - $\frac{2}{5}}$  ) * 250 = 50 more kids.

#3: If the girls to boys ratio at an elementary school is 3 to 4 and there are 123 girls, how many boys are there at that elementary school?
Solution:
This one is easy, you set up the equation. Just make sure the numbers line up nicely.
Let there be "x" boys. $\frac{3}{4}$ = $\frac{123}{x}$ . Cross multiply to get x. Or since 123 is 41 * 3;
4 * 41 = 164 boys

#4 : If the girls to boys ratio at an elementary school is 2 to 5 and there are 78 more boys than girls, how many girls are there at the elementary school?

Solution I:
Again, you can use the algebra and let there be 2x girls and 5x boys.
According to the given, 5x - 2x = 3x = 78 so x = 26
Plug in and you get there are 2*26 = 52 girls and 5*26 = 130 boys.

Solution II:
If you keep expanding the ratio using the same multiples, 2 : 5 = 4 : 10 : 6 : 15...
Do you see that the difference of boys and girls are always multiples of 3 ( 5 - 2  = 3).
$\frac{78}{3}$ = 26 so there are 2 * 26 or 52 girls