Ratio and Proportion

Scott Solomon

Ratio and Proportion     Ratio and Proportion     Ratio and Proportion


Ratio
Ratio refers to a comparison between two similar things.

Ratio can be represented with a colon.
a:b

Ratio can also be represented as a fraction.
\bf\displaystyle\frac{a}{b}

 


Ratio of yellow beads to green beads = 1 to 2 = 1:2 = \bf\displaystyle\frac{1}{2}

 


Ratio of yellow beads to green beads = 2 to 4 = 2:4 = \bf\displaystyle\frac{2}{4}

 


Ratio of yellow beads to green beads = 3 to 6 = 3:6 = \bf\displaystyle\frac{3}{6}

 


Proportion
Proportion refers to two equal ratios.
a:b = c:d
\bf\displaystyle\frac{a}{b} = \bf\displaystyle\frac{c}{d}

For ratios \bf\displaystyle\frac{a}{b} and \bf\displaystyle\frac{c}{d} to be in proportion → ad = bc

 


To Prove Ratios are in Proportion, Cross-Multiply
In order to cross-multiply, it is helpful to understand algebraic equations.
It is also helpful to know your multiplication and division tables.

\bf\displaystyle\frac{a}{b} = \bf\displaystyle\frac{c}{d}
For ratios \bf\displaystyle\frac{1}{2} and \bf\displaystyle\frac{6}{12} →
ad = 1 × 12
bc = 2 × 6
1 × 12 = 2 × 6
12 = 12
Because ad = bc, ratios \bf\displaystyle\frac{1}{2} and \bf\displaystyle\frac{6}{12} are in proportion.

 

To Solve for an Unknown Variable in Proportionate Ratios, Cross-Multiply
In order to cross-multiply, it is helpful to understand algebraic equations.
It is also helpful to know your multiplication and division tables.

\bf\displaystyle\frac{a}{b} = \bf\displaystyle\frac{c}{d}
For ratios \bf\displaystyle\frac{2}{4} and \bf\displaystyle\frac{5}{d} →
ad = bc
ad = 2 × d
bc = 4 × 5
2 × d = 4 × 5
2d = 20
\bf\displaystyle\frac{2d}{2} = \bf\displaystyle\frac{20}{2}
d = 10

 

Question
Prove that the ratios of yellow to green beads in the image below are in proportion.
   

Answer
\bf\displaystyle\frac{1}{2} = \bf\displaystyle\frac{2}{4}

Answer Process
\bf\displaystyle\frac{a}{b} = \bf\displaystyle\frac{c}{d}
For ratios \bf\displaystyle\frac{1}{2} and \bf\displaystyle\frac{2}{4} →
ad = 1 × 4
bc = 2 × 2
1 × 4 = 2 × 2
4 = 4
Because ad = bc, ratios \bf\displaystyle\frac{1}{2} and \bf\displaystyle\frac{2}{4} are in proportion.


Question
In the image below, two ramps are in proportion.  Find the length L of the larger ramp.

Answer
6

Answer Process
\bf\displaystyle\frac{a}{b} = \bf\displaystyle\frac{c}{d}
For ratios \bf\displaystyle\frac{2}{3} and \bf\displaystyle\frac{4}{L} →
ad = 2 × L
bc = 3 × 4
2 × L = 3 × 4
2L = 12
\bf\displaystyle\frac{2L}{2} = \bf\displaystyle\frac{12}{2}
L = 6


Question
In the image below, two ramps are in proportion.  Find the length Q of the smaller ramp.

Answer
10

Answer Process
\bf\displaystyle\frac{a}{b} = \bf\displaystyle\frac{c}{d}
For ratios \bf\displaystyle\frac{20}{40} and \bf\displaystyle\frac{5}{Q} →
ad = 20 × Q
bc = 40 × 5
20 × Q = 40 × 5
20Q = 200
\bf\displaystyle\frac{20Q}{20} = \bf\displaystyle\frac{200}{20}
Q = 10


Practice – Questions
1.  Prove that the ratios of yellow to green beads in the image below are in proportion.
     

 

2.  In the image below, two ramps are in proportion.  Find the length L of the larger ramp.

 

3.  In the image below, two ramps are in proportion.  Find the length Q of the smaller ramp.

 

4.  In the image below, two rectangular boxes are in proportion.  Find the length L of the larger box.

 

5.  In the image below, two rectangular boxes are in proportion.  Find the width W of the smaller box.


Practice – Answers
1.  For ratios \bf\displaystyle\frac{1}{2} and \bf\displaystyle\frac{3}{6} →
ad = 1 × 6
bc = 2 × 3
1 × 6 = 2 × 3
6 = 6
Because ad = bc, ratios \bf\displaystyle\frac{1}{2} and \bf\displaystyle\frac{3}{6} are in proportion.

2.  21

3.  6

4.  21

5.  3

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