 ## Step-by-step Guide: Word Problems of Comparing Ratios

Ratios are prevalent in our everyday life. It could be calculating the ratio of ingredients in a recipe or comparing rates of different products or services. Therefore, understanding how to solve word problems involving ratios is a valuable skill. This guide will break down the process into easy-to-follow steps.

### Step 1: Understand the Problem

Begin by carefully reading the problem. Understand what the problem is asking for and identify the ratios involved. The ratios could be presented in various forms such as fractions, words (e.g., 3 to 4), or with a colon (e.g., 3:4).

### Step 2: Write Down the Ratios

Once you have identified the ratios in the problem, write them down. It is generally easier to work with ratios when they are written as fractions. For example, if the problem presents the ratio 3:4, write it down as $$\frac{3}{4}$$.

### Step 3: Identify the Relationship

Determine whether the ratios are describing parts of a whole or the relationship between two different quantities. This will help you understand how to compare the ratios.

### Step 4: Compare the Ratios

To compare ratios, you can use cross-multiplication. Multiply the numerator of the first ratio by the denominator of the second ratio. Do the same for the denominator of the first ratio and the numerator of the second ratio. If the two products are equal, the ratios are equivalent.

### Step 5: Answer the Question

Based on your comparison, answer the question asked in the problem. This could involve determining which ratio is larger, whether the ratios are equivalent, or some other question related to the ratios.

### Conclusion

Understanding and comparing ratios is a crucial skill not just in mathematics, but in many areas of life. With practice, solving word problems involving ratios can become second nature. So keep practicing, and you'll master this skill in no time!

### Example

Suppose your friend Jack has a car that can travel 270 miles on 9 gallons of gas. Meanwhile, your car can travel 250 miles on 8 gallons of gas. Who has the car with better fuel efficiency, and by how much?

First, we need to find the miles-per-gallon ratios for both cars.

For Jack's car, the ratio is 270 miles to 9 gallons, or simply $$\frac{270}{9} = 30$$ miles per gallon.

For your car, the ratio is 250 miles to 8 gallons, or simply $$\frac{250}{8} = 31.25$$ miles per gallon.

Now it's clear that your car is more fuel efficient. It travels $$31.25 - 30 = 1.25$$ miles more per gallon than Jack's car.

So, the answer is your car is more fuel efficient by 1.25 miles per gallon.

### Exercises

1) Tom has 5 red marbles and 3 blue marbles in a bag, while Jane has 12 red marbles and 8 blue marbles. Whose bag has a greater ratio of red to blue marbles?

2) A recipe calls for 2 cups of flour for every 3 cups of sugar. Another recipe requires 4 cups of flour for every 6 cups of sugar. Which recipe has a larger ratio of flour to sugar?

3) A car travels $$60$$ miles in $$2$$ hours, and a train travels $$120$$ miles in $$3$$ hours. Which has a higher speed ratio (miles per hour)?

4) Jack reads $$4$$ books in $$6$$ months, and Jill reads $$8$$ books in $$12$$ months. Who reads more books per month?

5) An artist uses $$3$$ tubes of blue paint for every $$5$$ tubes of yellow paint for a project. Another artist uses $$9$$ tubes of blue paint for every $$15$$ tubes of yellow paint for a similar project. Who uses more blue paint compared to yellow paint?

6) A recipe requires $$3$$ eggs for $$2$$ cups of milk. How many eggs would be needed for $$4$$ cups of milk?

7) In a classroom, the ratio of boys to girls is $$3$$ to $$2$$. If there are $$15$$ boys, how many girls are there?

8) A car can travel $$75$$ miles with $$3$$ gallons of gas. How many miles can it travel with $$5$$ gallons of gas?

9) For every $$5$$ students in a class, there are $$3$$ computers. How many computers would be needed for a class of $$20$$ students?

10) The ratio of dogs to cats in a pet store is $$4:7$$. If there are $$28$$ cats, how many dogs are there in the pet store?

1) The ratio of red to blue marbles is the same for both Tom and Jane $$(5:3)$$ for Tom and $$(12:8)$$ for Jane, which reduces to $$(5:3)$$, so neither has a greater ratio of red to blue marbles.

2) The ratio of flour to sugar is the same for both recipes $$(2:3)$$ for the first recipe and $$(4:6)$$ for the second recipe, which reduces to $$(2:3)$$, so neither recipe has a larger ratio of flour to sugar.

3) The car has a speed ratio of $$30$$ miles per hour $$(60 \text{ miles} / 2 \text{ hours})$$ and the train has a speed ratio of $$40$$ miles per hour $$(120 \text{ miles} / 3 \text{ hours})$$, so the train has a higher speed ratio.

4) Both Jack and Jill read the same number of books per month $$(4/6 = 0.67)$$ books per month and Jill reads $$(8/12 = 0.67)$$ books per month).

5) Both artists use the same ratio of blue to yellow paint $$(3:5)$$ for the first artist and $$(9:15)$$ for the second artist, which reduces to $$(3:5)$$.

6) The ratio of eggs to milk in the recipe is $$(3:2)$$, so if $$4$$ cups of milk are used, $$6$$ eggs would be needed $$(4 \times 3/2 = 6)$$.

7) In the classroom, the ratio of boys to girls is $$(3:2)$$, so if there are $$15$$ boys, there would be $$10$$ girls $$(15 \times 2/3 = 10)$$.

8) The car can travel $$25$$ miles per gallon of gas $$(75 \text{ miles} / 3 \text{ gallons})$$, so with $$5$$ gallons of gas, it could travel $$125$$ miles $$(25 \text{ miles/gallon} \times 5 \text{ gallons} = 125 \text{ miles})$$.

9) For every $$5$$ students, there are $$3$$ computers, so for a class of $$20$$ students, $$12$$ computers would be needed $$(20 \text{ students} \times 3/5 = 12 \text{ computers})$$.

10) The ratio of dogs to cats in the pet store is $$(4:7)$$, so if there are $$28$$ cats, there would be $$16$$ dogs $$(28 \times 4/7 = 16)$$.

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