Determining Equivalent Ratios

Determining Equivalent Ratios exercise

• Explain what happens if we enlarge the bowl using ratios and ratio tables.

  Hints

  A ratio is reduced when there are no common factors. Here are some examples:

  • The ratio $4:5$ is reduced because there are no whole numbers other than $1$ that divide into both $4$ and $5$.

  • The ratio $7:1$ is reduced.

  • The ratio $30:12$ is not reduced because we can divide both values by $6$ to get $5:2$.

  • $5:2$ is reduced.

  We can multiply the values of a ratio to get an equivalent ratio. Here are some examples:

  • Given the ratio $4:7$, we can multiply by $3$ to get $12:21$. $12:21$ is equivalent to $4:7$ because $\frac{12}{21}=\frac47$.

  • Given the ratio $1:3$ we can multiply by $8$ to get the equivalent ratio $8:24$.

  • Given the ratio $4:9$ we can multiply by $\frac12$ to get the equivalent ratio $2:4.5$.

  Given two ratios, we can compare them by reducing:

  • The ratios $6:10$ and $3:5$ are equivalent. Dividing the numerator and denominator of $\frac{6}{10}$ by $2$ gives us $\frac35$.

  • The ratios $45:150$ and $6:20$ are equivalent. $\frac{45}{150}$ and $\frac{6}{20}$ both reduce to $\frac{3}{10}$.

  • The ratios $21:49$ and $9:24$ are not equivalent. $\frac{21}{49}=\frac37$ while $\frac{9}{24}=\frac38$.

  Solution

  The reduced ratio of the diameter of the bowl to its height can be written $8$ to $3$, $8:3$, or $\frac83$. If we enlarge the bowl by a factor of $2$, the ratio of diameter to height becomes $16:6$. This ratio is equivalent to $8:3$ because $\frac{16}{6}=\frac83$. In fact, all of the ratios in the table are equivalent because all of them reduce to $8:3$. If we multiply $8$ and $3$ by $10$, we get the ratio $80:30$, which is equivalent to $8:3$.

• Define various key words related to ratios.

  Hints

  A ratio can be written several ways. For example, the ratio $10:3$ can be written:

  • $10:3$
  • $10 ~\text{to}~ 3$
  • $\frac{10}{3}$

  Ratios must have positive numbers. Zero and negative numbers are excluded. The number $1$ is not excluded.

  • All the ratios in a ratio table are equivalent, otherwise we don't call it a ratio table.

  • A ratio in a table may already be completely reduced, in which case it cannot be reduced further.

  If we reverse the order of the ratio, we have changed its meaning. For example:

  • The ratio of the length of a car to its height is $17~\text{ft}:5~\text{ft}$.

  • A ratio of length to height of $5:17$ is very different. That would be a very tall car!

  • However, if we are careful to redefine $17:5$ as the ratio of height to length, then we are still talking about the original car.

  We are not required to write ratios with the largest number first, or to put them in a particular order. But when we choose an order for a ratio, we cannot change the order without changing the meaning. Here's the one exception to that rule:

  • The ratio $1:1$ is the same when it's reversed.

  Solution

  • Ratios are not always written with the largest number first.

  • Ratios in a ratio table are always equivalent.

  • We cannot create equivalent ratios by multiplying each value in the ratio by a different number.

  • Multiplying the values of a ratio by the same number results in an equivalent ratio.

  • Enlarging an object by multiplying its dimensions by the same value preserves the ratio of the dimensions.

  • All of the ratios in a ratio table are equivalent fractions.

  • We can find equivalent ratios by multiplying by a constant.

  • There are several ways to write a ratio.

  • A ratio table cannot feature negative numbers.

  • A ratio can have $1$ as one of its values.

  • A ratio cannot have $0$ as one of its values.

  • Not every ratio can be reduced (some are already reduced).

  • If a ratio is written in reverse order, it is often a different ratio. For example: $4:5\neq5:4$.

  • The ratio $1:1$ written in reverse order is the same as the original ratio.

• Solve the real-world problem using a ratio table.

  Hints

  To find an equivalent ratio from a given ratio, we can always reduce that ratio, then multiply by a factor to get the desired ratio. Here are some examples:

  $\begin{array}{c|c} \text{Sugar} & \text{Cinnamon}\\ \hline ? & 7\\ 6 & 14\\ 30 & ?\\ \end{array}$

  • In the above table, we can see that the given ratio $6:14$ can be reduced by dividing by $2$.

  • The resulting ratio is $3:7$.

  • Multiplying by $10$ gives us $30:70$.

  From a given ratio, we can always begin by trying to reduce it to a simpler ratio, then multiplying by a specific factor to get a desired ratio. Finding the correct factor may require division. Here's an example:

  $\begin{array}{c|c} \text{Apples} & \text{Bananas}\\ \hline 5 & ?\\ 15 & 33\\ 490 & ? \end{array}$

  • From $15:33$, our reduced ratio is $5:11$.

  • $490$ divided by $5$ gives us $98$.

  • So we multiply $11$ by $98$ to get $1078$.

  • The last ratio in the table is $490:1078$.

  Given a ratio, we may not need to reduce to create the desired equivalent ratio:

  $\begin{array}{c|c} \text{Shirts} & \text{Pants}\\ \hline 6 & 15\\ 12 & ?\\ \end{array}$

  • Although the ratio $6:15$ is not reduced in the above example, we can see that $12$ is $6$ times $2$.

  • So we multiply $15$ by $2$ to get $30$.

  • The last ratio in the table is $12:30$.

  Solution

  The ideal ratio of Strawberry bushes to Blueberry bushes in a garden is represented by the incomplete ratio table at right. The smallest garden is represented by the ratio $3:5$. If we want a bigger garden, we must increase the values of the ratio. If we want $12$ Strawberry bushes, we multiply the reduced ratio of $3:5$ by $4$. The resulting ratio will be $12:20$.

  Malala keeps track of her hours of Homework and Basketball practice per week. The ratio table shows the ideal relationship between the two. Malala wants to play Basketball for $18$ hours this week. How much time should she dedicate to Homework? The given ratio $4:6$ can be reduced to $2:3$. Multiplying this ratio by $6$ gives us $12:18$. So if Malala wants to play basketball for $18$ hours, she should do math homework for $12$ hours!

  The ideal ratio of Blue candies to Red candies is shown in the ratio table. Although the number of candies can increase, the ratio between Blue and Red stays the same. How many Blue candies are needed if we have $150$ Red candies? We don't always have to reduce a given ratio. The given ratio of $60:75$ can be multiplied directly by $2$ to get $120:150$. That means we need $120$ Blue candies to preserve the ideal ratio.

• Identify which ratios are equivalent to the ratios displayed in the central elements.

  Hints

  To compare two ratios, we can reduce them:

  $\begin{array}{c|c|c} \text{1st Fraction} & \text{2nd Fraction} & \text{Equivalent?}\\ \hline\ &&\\ \frac{6}{14}=\frac37 & \frac{15}{35}=\frac{3}{7} & \text{yes}\\ &&\\ \frac{2}{30}=\frac{1}{15} & \frac{30}{2}=\frac{15}{1} & \text{no}\\ &&\\ \frac{10}{12}=\frac56 & \frac{18}{24}=\frac34 & \text{no} \end{array}$

  We can multiply fractions by factors to create equivalent ratios. The ratio $A:B$ below is $3:7$. Multiplying this ratio by the given factors results in equivalent ratios as shown:

  $\begin{array}{r|c|c} \text{Multiply} & \text{A} & \text{B}\\ \hline & 3 & 7\\ \times2 & 6 & 14\\ \times3 & 9 & 21\\ \times5 & 15 &35\\ \times10 & 30 & 70 \end{array}$

  Sometimes we can go directly from a given ratio that isn't reduced, to another equivalent ratio, as in the example below:

  $\begin{array}{c|c} \text{A} & \text{B}\\ \hline 40 & 25\\ ? & 150\\ \end{array}$

  • $25\times6=150$.
  • So we can compute $40\times6=240$.
  • The resulting ratio is $240:150$.

  Solution

  One way to simplify this problem is to reduce the ratios for each of the central ratios first. Then focus on those reduced ratios as you are matching the remaining ratios.

  $\begin{array}{r|r|r|r} \text{Central Ratio} \rightarrow & 5:12 & 12:8=3:2 & 8:16=1:2\\ \hline & 15:36=5:12 & 6:4=3:2 & 4:8=1:2\\ & 10:24=5:12 & 21:14=3:2 & 200:400=1:2 \\ & 50:120=5:12 & &\\ \end{array}$

• Decide which fractions are equivalent to each other.

  Hints

  To reduce a fraction:

  • Look for a common factor in the numerator and denominator.

  • Divide the numerator and denominator by the common factor.

  • Repeat this process until there are no more common factors.

  Here are some examples:

  $\begin{array}{c|c|c} \text{Fraction} & \text{Common Factor} & \text{Reduced Result}\\ \hline\ &&\\ \frac{6}{15} & 3 & \frac25\\ &&\\ \frac{21}{42} & 21 & \frac12\\ &&\\ \frac{40}{32} & 8 & \frac54 \end{array}$

  Two fractions can be equivalent even though neither of them is reduced. Reduce each fraction and compare their results to determine if they are equivalent. Here are some examples:

  \begin{array}{c|c|c} \text{1st Fraction} & \text{2nd Fraction} & \text{Equivalent?}\\ \hline\ &&\\ \frac{12}{21}=\frac47 & \frac{24}{42}=\frac{4}{7} & \text{yes}\\ &&\\ \frac{12}{15}=\frac45 & \frac{15}{18}=\frac56 & \text{no}\\ &&\\ \frac{10}{30}=\frac13 & \frac{30}{60}=\frac12 & \text{no} \end{array}

  A whole number can be written as a fraction. $9=\frac91$.

  Solution

  Here is a table showing reducing to find the equivalent fractions.

• Complete the Ratio Tables.

  Hints

  We can reduce a given ratio by dividing by a common factor of each value:

  $\begin{array}{r|c|l} \text{Given Ratio} & \text{Common Factor} & \text{Reduced Ratio}\\ \hline 22:11 & \div11= & 2:1\\ 39:26 & \div13= & 3:2\\ 50:75 & \div25= & 2:3 \end{array}$

  We can find equivalent ratios by multiplying to get the desired result.

  $\begin{array}{r|c|c} \text{Multiply} & \text{A} & \text{B}\\ \hline & 5 & 3\\ \times2 & 10 & 6\\ \times3 & 15 & 9\\ \times5 & 25 &15\\ \times10 & 50 & 30 \end{array}$

  Sometimes it can be helpful to divide in order to determine what to multiply the reduced ratio by:

  $\begin{array}{c|c} \text{A} & \text{B}\\ \hline 5 & 17\\ ? & 102\\ \end{array}$

  • $\frac{102}{17}=6$.
  • This means we multiply $5\times6=30$ to complete the second ratio.
  • The second ratio is $30:102$.

  Solution

  $\begin{array}{c|c|c} 3:2 & 7:4 & 8:9\\ 15:10 & 21:12 & 40:45\\ 27:18 & 35:20 & 32:36\\ \end{array}$