Solving Problems with Equivalent Ratios

Solving Problems with Equivalent Ratios exercise

• Use tape diagrams and equivalent ratios to solve problems.

  Hints

  Given the ratio between different quantities and the total number of those quantities combined, we can use tape diagrams to find out how much of each quantity we have.

  For example:

  • Suppose we are given the ratio of Peanuts to Walnuts: $P:W=3:2$.
  • We are also given a total of $P+W=30$ nuts in total.
  • We know there are $3+2=5$ rectangles in total.
  • That means each rectangle represents $\frac{30}{5}=6$ nuts.
  • Since Peanuts have $3$ rectangles, there are $3\times6=18$ Peanuts.
  • Since Walnuts have $2$ rectangles, there are $2\times6=12$ Walnuts.

  We can use tape diagrams to solve problems involving three quantities. Suppose that in a restaurant, the ratio of Chairs to Tables to Stools is $3:2:1$. And suppose there are $42$ pieces of furniture total. How many of each type of furniture are in the restaurant?

  • There are $6$ total rectangles.
  • $\frac{42}{6}$ gives us $7$ pieces of furniture per rectangle.
  • The number of Chairs is $3\times7=21$.
  • The number of Tables is $2\times7=14$.
  • The number of Stools is $1\times7=7$.
  • We can check our answer by adding: $21+14+7=42$ pieces of furniture.

  Given a ratio of quantities and a total number of those quantities:

  • Draw rectangles to represent the ratio between different quantities.
  • Find the total number of rectangles.
  • Divide the total of the quantities by the total rectangles to get the quantity per rectangle.
  • Then multiply to find how much of each quantity you've got.

  Solution

  Because the ratio $P:C:D$ is $5:2:3$ , Phalaenopsis gets $5$ rectangles, Cattleya gets $2$, and Dendrobium gets $3$. There are $10$ rectangles in total. Because there are $100$ orchids in the shop, we divide $100$ by $10$ to give us $10$ orchids per rectangle. Because Phalaenopsis has $5$ rectangles, there are $5\times10=50$ plants of this type. Because Cattleya has $2$ rectangles, there are $2\times10=20$ plants of this type. Dendrobium has $3$ rectangles and so there are $3\times10=30$ plants of this type.

• Use tape diagrams to solve the given ratio problem.

  Hints

  Given a ratio, we can determine how many more rectangles are assigned to one quantity over another.

  For example:

  • Suppose we are give that the ratio of Cars to Trucks at a car dealership is $C:R=14:11$.
  • We are also given that there are $12$ more Cars than Trucks.
  • Cars have $14$ rectangles and Trucks have $11$.
  • Cars have $14-11=3$ more rectangles than Trucks.
  • Because there are $12$ more Cars than Trucks and that's represented by $3$ rectangles, each rectangle represents $\frac{12}{3}=4$ automobiles.
  • So there are $14\times4=56$ Cars and $11\times4=44$ Trucks.
  • We can confirm our answer by subtracting the number of Trucks from the number of Cars: $56-44=12$.

  Given a ratio and the difference between the quantities, we can find the amount of each quantity:

  • Assign rectangles to each quantity based on the given ratio.
  • Subtract the number of rectangles of one quantity from that of the other.
  • Divide the given difference between the quantities by the number of rectangles from the last step. This gives you the quantity per rectangle.
  • Multiply the number of rectangles for each quantity times the quantity per rectangle to get the value for each quantity.

  Suppose the ratio of a quantity $A$ to a quantity $B$ is $8:5$, and the difference $A-B=30$. What's quantity of $A$ and the quantity of $B$?

  • $8-5=3$.
  • $\frac{30}{3}=10$.
  • The quantity of $A$ is $8\times10=80$.
  • The quantity of $B$ is $5\times10=50$.
  • We can check our answer: $80-50=30$, the difference given in the original problem.

  Solution

  Given:

  • The ratio of Blueberries to Raspberries is $4$ to $7$.
  • There are $27$ more Raspberries than Blueberries.

  Solution:

  • Write the ratio in proper notation:
  • $B:R=4:7$.
  • Subtract to find the difference in the number of rectangles assigned to each quantity:
  • $7-4=3$.
  • Divide to find the number of berries per rectangle:
  • $\frac{27}{3}=9$ berries per rectangle.
  • Multiply to find the number of Blueberries and Raspberries:
  • $4\times9=36$ Blueberries, and $7\times9=63$ Raspberries.
  • Subtract these values to check your answer:
  • $63-36=27$.

• Determine the total number of Venus fly traps and roses in Paul's shop.

  Hints

  Suppose the ratio of Road Bikes to Mountain Bikes in a bike shop is $R:M=5:7$, and there are 12 more Mountain bikes than Road bikes in the shop.

  • The tape diagram will show $2$ more rectangles for $M$ than $R$.
  • Those $2$ rectangles represent $12$ bikes.
  • So each rectangle represents $\frac{12}{2}=6$ bikes.
  • Because we have $5+7=12$ rectangles, we have $12\times6=72$ bikes in total.

  Suppose the ratio of Burritos to Tacos at a Mexican food truck is given as $B:T=5:9$, and there are $20$ more Tacos than Burritos.

  • The tape diagram will show $4$ more rectangles for $T$ than $B$.
  • Those $4$ rectangles represent $20$ food items.
  • So each rectangle represents $\frac{20}{4}=5$ food items.
  • Because we have $5+9=14$ rectangles, we have $14\times5=70$ food items in total.

  Solution

  At Paul’s plant shop there are $5$ Venus Fly Traps for every $8$ Roses. Paul has $90$ more Roses than Venus Fly Traps. How many Venus Fly Traps and Roses does Paul’s plant shop have?

  • The ratio of Venus Fly Traps to Roses is given as $V:R=5:8$.
  • The tape diagram has $8$ rectangles on top and $5$ on the bottom, so the top represents Roses and the bottom represents Venus Fly Traps.
  • There are $3$ more rectangles for Roses than for Venus Fly Traps.
  • Those $3$ rectangles represent $90$ plants.
  • So each rectangle represents $\frac{90}{3}=30$ plants.
  • There are $5+8=13$ rectangles in total, so there are $13\times30=390$ plants in total.

• Calculate the solution to the following problems using tape diagrams.

  Hints

  Suppose the ratio of Planets to Moons around a star is $11:14$. There are $100$ celestial bodies in total associated with the star. How many more Moons than Planets are there?

  • All the celestial bodies are represented by $25$ rectangles.
  • Each rectangle represents $\frac{100}{25}=4$ celestial bodies.
  • There are $14-11=3$ more rectangles for Moons than for Planets.
  • Those $3$ rectangles represent $3\times4=12$ Moons.

  Suppose the ratio of Football teams to Basketball teams in a town is $4$ to $5$. There are $3$ more Basketball teams than Football teams. How many teams are there in total?

  • The difference in rectangles is $1$.
  • This rectangle represents $3$ teams.
  • There are $4+5=9$ rectangles in total.
  • There are $9\times3=27$ teams in total.

  Suppose the ratio of Plates to Cups in a kitchen is $7$ to $10$, and there are $12$ more Cups than Plates. How many Plates are there?

  • There are $10-7=3$ more rectangles for Cups than Plates.
  • Those $3$ rectangles represent $12$ pieces of dinnerware.
  • Each rectangle represents $\frac{12}{3}=4$ pieces of dinnerware.
  • There are $7 \times 4=28$ Plates.

  Solution

  A clothing store features Shirts and Pants in the ratio of $3:6$. There are $42$ more Shirts than pairs of Pants. How many articles of clothing in total are in the store?

  • $6-3=3$ more rectangles for Shirts than Pants.
  • $\frac{42}{3}=14$ articles of clothing per rectangle.
  • $3+6=9$ rectangles in total.
  • $9\times 14 =126$ articles of clothing in the store.

  An art museum features Cubist and Impressionist works in the ratio $11$ to $7$. There are $24$ fewer Impressionist works than Cubist works. How many Impressionist works are featured at the museum?

  • $11-7=4$ more rectangles for Cubist works.
  • $\frac{24}{4}=6$ works of art per rectangle.
  • $7 \times 6 = 42$ Impressionist works.

  A packet of trail mix features a ratio of Peanuts to Raisins of $9:10$. There are $2$ more Raisins than Peanuts in the packet. How many Raisins are in the packet?

  • There is $1$ more rectangle for Raisins than Peanuts.
  • This rectangle represents $2$ Raisins.
  • There are $10\times2=20$ Raisins in total.

  A music playlist features songs from the Eighties and Nineties in the ratio of $9:7$. If there are $32$ songs in the playlist, how many more songs from the Eighties are there?

  • There are $9+7=16$ rectangles in total.
  • These rectangles represent $32$ songs in total.
  • Each rectangle represents $\frac{32}{16}=2$ songs.
  • There are $9-7=2$ more rectangles for the Eighties than the Nineties.
  • That represents $2\times2=4$ songs.

• Create a tape diagram to represent the given ratios.

  Hints

  If the ratio $A:B=25:10$, we can reduce this ratio by dividing each value by $5$:

  • $\frac{25}{5}=5$
  • $\frac{10}{5}=2$

  Therefore, the reduced ratio is $A:B=5:2$.

  If the ratio $A:B=7:3$, we can say that the ratio $B:A=3:7$.

  Given the following ratios, we can combine them into a single ratio statement involving three values:

  • $A:B=5:7$
  • $B:C=7:4$
  • Therefore: $A:B:C=5:7:4$

  We can represent this with a rows of $5$, $7$, and $4$ rectangles.

  Solution

  The given ratios are $P:M=12:8$ and $T:M=15:30$.

  • $P:M=12:8$ can be reduced.
  • Dividing each value by $4$ gives us $P:M=3:2$.
  • $T:M=15:30$ can also be reduced.
  • Dividing each value by $2$ gives us $T:M=1:2$.
  • If $T:M=1:2$, then $M:T=2:1$.
  • Therefore, $P:M:T=3:2:1$.
  • In the tape diagram, $P$ gets $3$ rectangles, $M$ gets $2$, and $T$ gets $1$.

• Confirm or correct the solution to each word problem.

  Hints

  Remember, with ratios, order matters. If the ratio of Cups to Bowls is $3:10$, we must be careful to assign $3$ rectangles to Cups, and $10$ rectangles to Bowls, not the other way around.

  Remember, we have used two different methods to find the quantity per rectangle:

  • Given $A:B=2:7$,
  • And there are $45$ more of $A$ than $B$, we compute:
  • $\frac{45}{5}=9$ units per rectangle.

  • Given $A:B=2:7$,
  • And there are $45$ units in total, we compute:
  • $\frac{45}{9}=5$ units per rectangle.

  If we can find the values of two quantities, we can compute a reduced ratio between them. For example:

  • Given $A=400$ and $B=1400$,
  • $A:B=400:1400$.
  • We can reduce this by dividing each value by $200$:
  • $A:B=2:7$

  Solution

  A scout troop is selling boxes of cookies. The ratio of boxes of Chocolate Chip to boxes of Sugar cookies is $5$ to $6$. There are $33$ boxes of cookies in total. Chocolate Chip cookies are represented by $5$ rectangles, and Sugar by $6$ rectangles. There are $\frac{33}{11}=3$ boxes per rectangle. This gives us $5\times 3=15$ boxes of Chocolate Chip, and $6\times 3=18$ boxes of Sugar cookies.

  A grove of trees has Spruce trees and Fir trees such that $S:F=6:3$. There are $90$ more Spruce trees than Fir trees. There are $3$ more rectangles for Spruce than for Fir. So there are $\frac{90}{3}=30$ trees per rectangle. That means there are $6\times30=180$ Spruce trees and $3\times30=90$ Fir trees.

  A fruit table at a marathon features Apples, Oranges, and Bananas. There are $48$ more Bananas than Oranges. The ratio $O:B=6:8$. There are $264$ Apples and Oranges. What is the ratio of Apples to Oranges?

  There are $2$ more rectangles for Bananas than for Oranges. This means there are $\frac{48}{2}=24$ pieces of fruit per rectangle. There are $6$ rectangles for Oranges, so there are $6\times 24=144$ Oranges. Since there are $264$ Apples and Oranges, there must be $264-144=120$ Apples. The ratio of Apples to Oranges is $120:144$ which reduces to $5:6$.