Finding a total using ratios

On the homework there is a problem where you have to use a ratio to find the yards each person ran, with the total yards ran being 150.  Here is an example of how to do that.

Scott is stronger than Shawn.  They have a goal to lift 500 pounds together.  Scott is 1 1/2 time stronger than Shawn, meaning for every 3 pounds that Scott can lift, Shawn can lift 2.  The ratio of their lifts are 3 to 2, or 3/2 as a fraction.  Here is how to solve who lifts how much weight.


When we add their weights lifted together, we need to reach 500 pounds.  If they were to lift the same about, that would be 250 pounds, or a 1 to 1 ratio.  As it is, for every 3 pounds Scott lifts, Shawn can lift 2.     This is a total of 5 lbs. We have to use an equivalent ratio.  If we multiply this by 100, they would lift 500 pounds.

3 x 100  = 300
2 x 100     200

CBP Quiz B Study Guide

Negative Numbers

We have been working on positive and negative numbers in terms of fractions on a number.  This video may help you and your student to understand this standard a little better.

Go to www.learnzillion.com

enter quick code  LZ1137

If you can't get to the video, here is a review.  

Positive numbers always fall to the right of 0 in ascending order, 1, 2, 3, 4, etc.

Negative numbers always fall to the left of 0, falling from -1, -2, -3, -4.

Positive numbers we are more familiar with.  The larger number is FARTHER from 0.  For example, 4 is farther from 0 and 1 is, so 4 is greater.  We do this by subtracting.  1-0=1  which is the distance from 0.  4-0= 4, which is it's distance from 0.  4 has a greater magnitude therefore 4 is greater.

Negative numbers have a different set of rules.  The number CLOSER to 0 is the great number.  -1 is greater than -4.  0- (-4) = 4, which is farther from 0. 0 - (-1) = 1, which is closer to 0 meaning it that is greater than -1.  

When adding and subtracting negative numbers, if we have a subtraction sign and a negative sign next to each other, we have to use parentheses to separate.  In math, we cannot have two signs touching, they must always be separated by a number.  One way to do this is to change the sign accordingly.  Here is the rule:

+ + = +
+ - = -
- + = -
- - = +

"Like, add, write the common sign
Unlike, subtract, keep the sign of the largerrrrr"

For example, we never write 4 + (+5), because we know it is understand that is is 4 + 5.

We do write this 4 + (- 5 ) really means 4 - 5.

The opposite holds true.  4 - (+ 5) really means 4 - 5.

This is true also.  4 - (-5) really means 4 + 5.

To get even more technical, the - in front of the () is actually a -1, so this means

4 -1(-5).  Order of Operations says to multiply the -1 by the -5, and a negative times a negative is always positive.

Rate Tables

Go to www.learnzillion.com and enter this code: LZ841  This will be helpful.

Let's look at a cost of an item.  If 48 oz juice bottle costs $4.00, what is the price per ounce?  How much would other amounts be?  Let's use a table.

Ounces	48	12	1	24	96	72	480
Price	$4

This means that for every 48 oz I will pay $4.  We can fill in the rest by using equivalent ratios.

Notice that 48 is larger than 4 and on the top row.  This means that no number on the bottom row should be larger than the one on the top.  Now to fill in the blanks, lets use equivalent fractions.

48      12

---- =  -----

4        

I do not know what the bottom number is.  However, I can use factors or fact families to determine what I need to do to get to 12 from 48.  I know:

12 x 4 = 48

4 x 12 = 48

So I also know:

48 ÷ 12 = 4

48 ÷ 4 = 12

SOOOOOO……

48 ÷ 4        12

-----      =   -----

4 ÷ 4           1

So 12 ounces is equal to $1.

Ounces	48	12	1	24	96	72	480
Price	$4	$1

The same principle applies to all of the numbers.

48        1

---- =  -----

4        

What do I need to get to 1 from 48?  Divide the top and bottom by 48!

48 ÷ 48           1                1

-----      =      -----     =      -----

4 ÷ 48           4/48             0.0833

So we would round to $0.08, or 8 cents.

Ounces	48	12	1	24	96	72	480
Price	$4	$1	0.08

The completed table:

Ounces	48	12	1	24	96	72	480	144
Price	$4	$1	$0.08	$2	$8	$6	$40	$12

Unit Rates

Unit rates are a ratio comparing one unit of one item to a quantity of another.  The most common unit rate is miles per hour.  The interstate speed limit is 70 miles per hour which literally means (barring any changes of speed due to accidents, getting off the interstate, bad traffic etc) that you would travel 70 miles in 1 hour.  This is usually represented as a ratio:

70 miles to 1 hour

70 miles : 1 hour

70 miles
1 hour

In class, we have been going over unit rates in terms of chewy gummy worms.  Let's say we have a giant gummy worm that is multi-colored.  Each color represents a segment.  Let's say we have a 4 segment chewy gummy worm and it looks like this:

If we divide this worm for 4 people, each person would get 1 segment, so the unit rate would be 1 segment per person.  If we only had two people, each person would get 2 segments, so the unit rate would be 2 segments per person.  But what if we have 8 people?

Here, we have to cut each segment in half, meaning each person would get 1/2 segment.  The unit rate now would be 1/2 segment per person.  This can be done a different way, but the ultimate goal would be to reach the 1/2 segment per person ratio (unit rate).

You can do this without drawing a picture.  Just divide the amount of segments by the number of people.  Below are two examples.

If we have an 8 segment worm divided among 6 people, you can set it up like a fraction (see picture).  We have 1 6 inside of 8, so we pull it out.  That leaves us with 2/6 left, which reduces to 1/3.  So the unit rate would be 1 1/3 segments per person.  What if the you have more people than segments?

Take 8 segments and divide it by 12 people.  Basically, simplify the fraction 8/12 by dividing both the numerator and the denominator by the GCF (4).  This give you 2/3.  The unit rate would be 2/3 of a segment per person.

The key when doing unit rates by fractions is to make what you want to be the 1 on the bottom (in this case we wanted PER PERSON).  Then simplify the fraction or divide.

Equivalent Fractions

We can find equivalent fractions by multiplying the numerator and denominator by a common factor.  This gives us multiples of the numerator and denominator.  We can continue to do this until we have the required number of equivalent fractions or until we reach the fraction we need.  The best idea is to stick with prime numbers or numbers that are easy to multiply by.  See the example below.