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A ratio shows the relationship between two quantities. In some cases, more than one ratio can represent the same relationship. This lesson will introduce *ratio tables* and explore how they can be used to represent relationships.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Take a look at these ratios and see how they can be simplified down to their simplest form.

What conclusions can be made about the ratio pairs?

A ratio is a comparison of two quantities that describes how much of one thing there is compared to another. Ratios are commonly represented using colon notation or as fractions. They are read as the ratio of $a$ to $b,$

where $b$ is a non-zero number.

$Fractionba Colon Notationa:b $

$Tearrik’s Ratio1527 Zain’s Ratio2545 $

These ratios can be simplified by finding the greatest common factor of their numerator and denominator. That factor can then be used to rewrite each ratio. Fraction Form | Greatest Common Factor | Rewrite | Simplify | |
---|---|---|---|---|

Tearrik | $1527 $ | $GCF(27,15)=3$ | $5⋅3 9⋅3 $ | $59 $ |

Zain | $2545 $ | $GCF(45,25)=5$ | $5⋅5 9⋅5 $ | $59 $ |

Scaling is a way of creating equivalent ratios by multiplying or dividing the numerator and denominator of a ratio by the same number. Sometimes it is necessary to scale back and forward to find an equivalent ratio for a particular situation. Consider the following scenario.

Chocolate bars are on sale at $14$ for $$6.$ This ratio can be scaled up and down to find the cost of $21$ chocolate bars.

$614 $ | |
---|---|

Scaling Back | Scaling Forward |

$6÷214÷2 $ | $3×37×3 $ |

$37 $ | $921 $ |

Determine whether the given pair of ratios is equivalent.

Mark loves taking pictures with his camera. One of his favorite shots is of the Walter Pyramid. The picture has a height-to-width ratio of $3:4.$

He wants to print it out and hang it on his bedroom wall, but he wants to print it in two different sizes equivalent to the original one.

a For the first size option, Mark would like to enlarge the original image four times. What will be the ratio of the height to the width of the new image? Write the answer in fraction form.

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b Mark needs to increase the size of his image by a factor of seven for the second print. What ratio represents the height-to-width ratio of this image? Answer in fraction form.

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a Write the original image's ratio in fraction form. Scale the original ratio by multiplying the numerator and the denominator by $4.$

b Multiply the original ratio by $7$ to get the ratio for the second size.

a Start by writing the size ratio of the original image as a fraction.

$3in.:4in.⇔4in.3in. $

Next, scale this ratio to find the ratio for the first size that Mark wants. Remember, he wants to enlarge the original image $four$ times. This means he should multiply the numerator and the denominator of the original ratio by $4$ to get the ratio for the new image.
$4in.×43in.×4 =16in.12in. $

The size of the first print will be $12$ inches in height and $16$ inches in width.
b Follow a similar process as in Part A to find the size ratio for the second print. This time, Mark wants to increase the size of his image by a factor of $seven.$ Multiply the numerator and denominator of the original ratio by $7$ to get the size ratio for the second image.

$4in.×73in.×7 =28in.21in. $

The second print will be $21$ inches in height and $28$ inches in width. Mark printed out his favorite photo of the Walter Pyramid in two sizes and hung them on his bedroom wall. Now his bedroom looks fantastic with the art!
Tables of equivalent ratios are called ratio tables. Consider the ratio table containing the distance Zain runs and the time it takes him to cover that distance.

Distance Run (Miles) | $1$ | $2$ | $3$ | $4$ |
---|---|---|---|---|

Time (Minutes) | $10$ | $20$ | $30$ | $40$ |

The ratios $101 ,$ $202 ,$ $303 ,$ and $404 $ are equivalent since each of them simplifies to a ratio of $101 .$ A ratio table can be created by multiplying or dividing each quantity in a ratio by the same number, or by repeatedly adding or subtracting the ratio.

Mark received a new phone from his parents that has an excellent camera for his photography hobby.

The aspect ratio, which represents the relationship between height and width, of the main camera is $2:3.$ Mark created a ratio table to explore the equivalent ratios that his photos can have.

Height | $2$ | $A$ | $6$ | $18$ |
---|---|---|---|---|

Width | $3$ | $6$ | $9$ | $B$ |

However, the table is missing some values.

a What are the values of $A$ and $B$ that complete the ratio table Mark created?

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b Which of the following graphs represents the ratio table for the aspect ratio of Mark's new phone camera?

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a A ratio table can be created by multiplying or dividing each quantity in a ratio by the same number or by adding or subtracting in increments of the ratio.

b Graph the ratio table created in Part A on a coordinate plane. The $x-$axis will represent the heights and the $y-$axis, the widths.

a There are two methods to find ratios for a ratio table. The first method is to multiply or divide both quantities in a ratio by the same number. The second method is to add and subtract quantities in increments of the original ratio. With this in mind, consider the given ratio table.

Height | $2$ | $A$ | $6$ | $18$ |
---|---|---|---|---|

Width | $3$ | $6$ | $9$ | $B$ |

Use the initial ratio of $2:3$ to determine the ratio for the second column. Note that adding $3$ to the width of the first ratio gives $6.$ This means that $2$ can be added to the height of the first ratio to find the height that corresponds to a width of $6,$ which is the value of $A.$

Use the ratio of $6:9$ to determine the corresponding width for a height of $18.$ Since $6⋅3=18,$ multiply $9$ by $3$ to get the value of $B.$

Therefore, $A=4$ and $B=27.$

b Graph the ratio table found in Part A on a coordinate plane. The $x-$axis will represent the heights and the $y-$axis the widths. Consider the ratio table from Part A.

Height, $x$ | $2$ | $4$ | $6$ | $18$ |
---|---|---|---|---|

Width, $y$ | $3$ | $6$ | $9$ | $27$ |

Because the table contains only positive values, only the first quadrant is required. Begin at the origin and move along the $x-$axis according to the number of units specified by the height. Then, move vertically the number of units specified by the width to graph each ratio.

This graph corresponds to option **A**.

Mark and Maya met up to compare the results of a five-day photo challenge. They each recorded the total number photos taken on each day in a table so that they could make a collage after finishing the challenge. The following table shows the data recorded by Maya and Mark.

Maya | Mark | ||
---|---|---|---|

Days | Photos | Days | Photos |

$1$ | $5$ | $1$ | $7$ |

$2$ | $10$ | $2$ | $14$ |

$3$ | $15$ | $3$ | $21$ |

$4$ | $20$ | $4$ | $28$ |

$5$ | $25$ | $5$ | $35$ |

a Choose the graph that accurately displays the information recorded in the tables.

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b Using both the table and the graph to determine which of the statements below are true. Select all that apply.

{"type":"multichoice","form":{"alts":["The ratio of photos to days for Mark is <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">7<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">:<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span>","The ratio of photos to days for Maya is <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">5<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">:<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span>","The ratio of days to photos for Mark is <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">7<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">:<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span>","The ratio of days to photos for Maya is <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">5<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">:<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span>","The line for Mark's data is steeper than the line for Maya.","The line for Maya's data is steeper than the line for Mark."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":[0,1,4]}

a Graph both tables on a coordinate plane. Begin at the origin and move along the $x-$axis by the number of units specified by the day. Next, move vertically the number of units specified by the photos taken to graph each data.

b Use the first column of each table to compare both ratios. Draw a line through each data. Which line is steeper?

a The tables will be graphed one by one on a coordinate plane. The $x-$axis will indicate the days, while the $y-$axis will show the number of photos taken. Begin by writing the ordered pair $(x,y)$ for each column of the table recorded by Maya.

Maya | ||
---|---|---|

Days | Photos | $(x,y)$ |

$1$ | $5$ | $(1,5)$ |

$2$ | $10$ | $(2,10)$ |

$3$ | $15$ | $(3,15)$ |

$4$ | $20$ | $(4,20)$ |

$5$ | $25$ | $(5,25)$ |

To graph the data, begin at the origin and move along the $x-$axis the number of units specified by the day. Then, move the number of units specified by the photos taken vertically to graph each ordered pair.

Before adding Mark's data to the graph, write each column of the table as an ordered pair.

Mark | ||
---|---|---|

Days | Photos | $(x,y)$ |

$1$ | $7$ | $(1,7)$ |

$2$ | $14$ | $(2,14)$ |

$3$ | $21$ | $(3,21)$ |

$4$ | $28$ | $(4,28)$ |

$5$ | $35$ | $(5,35)$ |

Each ordered pair can now be added to the graph.

The graph corresponds to the one given in option **B**.

b In this part, the ratios of photos taken to days passed for Mark and Maya will be compared. Consider the first column of both tables to see their unit ratios.

Maya | Mark | |
---|---|---|

Day, $x$ | $1$ | $1$ |

Photos, $y$ | $5$ | $7$ |

Maya's photo-to-day ratio is $5:1$ and Mark's is $7:1.$ Now, look at the graph drawn in Part A.

Notice that both sets of points appear to fit in straight lines. However, the line for Mark is steeper than the line for Maya. The statements that are true about the ratios of photos taken to days passed have been identified.

Statement | True? |
---|---|

The ratio of photos to days for Maya is $5:1.$ | $✓$ |

The ratio of photos to days for Mark is $7:1.$ | $✓$ |

The line for Mark's data is steeper than the line for Maya's. | $✓$ |

The ratio of days to photos for Maya is $5:1.$ | $×$ |

The ratio of days to photos for Mark is $7:1.$ | $×$ |

The line for Maya's data is steeper than the line for Mark's. | $×$ |

Maya suggested Mark post his outstanding artwork on a website for people to buy. As a beginner seller, Mark received a graph that showed how much he would earn per photo sold.

However, the graph does not provide information on Mark's earnings if he sells six photos. Assuming the ratio of photos sold to earnings remains constant, how much money would Mark earn from selling six photos?{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mord\">$<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["9"]}}

Use the relationship between a pair of consecutive points on the graph to determine the ratio of money earned to photos sold. Apply that ratio to find the point for selling six photos on the graph.

Mark wants to know how much money he would earn for selling six photos. It is given that the ratio of money earned per photo sold remains constant. Consider the graph carefully. The points appear to fall in a straight line.

The point $(4,6)$ is $2$ units to the right and $3$ units up from point $(2,3).$

This means that the ratio of the money earned to photos sold is $3:2$ — Mark will earn $$3$ for every $2$ photos sold. The amount of money he will earn by selling six photos can be determined using this ratio. Move $2$ units to the right and $3$ units up from $(4,2)$ to locate this point.

Therefore, Mark would earn $$9$ for selling six photos.

This lesson covered some uses of equivalent ratios and how ratio tables can be used to find them. Equivalent ratios are particularly useful when scaling ratios to maintain proportions. Take a look at a ratio table that shows how many photos Maya's printer can print per minute.

Photos | $4$ | $8$ | $12$ |
---|---|---|---|

Time (minutes) | $5$ | $10$ | $15$ |

Now consider the ratio table that shows the same information for Mark's printer.

Photos | $20$ | $24$ | $28$ |
---|---|---|---|

Time (minutes) | $25$ | $30$ | $35$ |

$25÷520÷5 =54 $

This means that the ratios $20:25$ and $4:5$ are the equivalent, which implies that all the ratios in both tables are also equivalent. A question that arises here is what would happen if these two ratio tables were plotted on a coordinate plane.
Notice that both data sets fit on the same line. This indicates that they are collinear. In conclusion, if two ratio tables with equivalent initial ratios are graphed, they will lie on the same line, meaning they are collinear.