# The Enchanted Forest of How to Compare Ratios – A Tale of Mathematical Adventure

Once upon a time, in a land far away where numbers and ratios danced harmoniously together, there existed fascinating challenges known as word problems. Let's embark on a mathematical adventure through the enchanted forest of comparing ratios, where we'll decode the secrets held within these word problems!

## 1. The Quest Begins: Understanding Word Problems of Comparing Ratios

Our adventure begins with understanding what these magical word problems of comparing ratios are. A ratio is a comparison of two quantities. For instance, if a cauldron of magical potion requires $$3$$ drops of moon dew to $$2$$ petals of a starflower, the ratio is $$3:2$$. When you encounter a problem that asks you to compare ratios, it’s like being asked to compare two different magical potion recipes!

## 2. Mastering the Magical Artefact: A Guide to Comparing Ratios

### A Wizard’s Guide to Comparing Ratios in Word Problems

#### Step 1: Decode the Word Problem

Our first task is to decipher what the problem is asking. Identify the two ratios and understand what each part of the ratio represents.

#### Step 2: Write Down the Ratios

Now that we understand our challenge, let’s express these ratios using the magical symbol “:”, separating the numbers that are being compared.

#### Step 3: Simplify the Ratios

To compare our ratios effectively, we need to simplify them. Just like simplifying a spell, this process ensures that we have the clearest and simplest form of our ratios.

#### Step 4: Compare the Ratios

Once the ratios are simplified, we can finally compare them. Are they the same, or different?

Let’s illustrate this with an example from the enchanted forest: A blue sprite can gather $$10$$ starflowers in $$2$$ hours, while a red sprite can gather $$15$$ starflowers in $$3$$ hours. Which sprite gathers starflowers at a faster rate?

1. Decode the Word Problem: We are comparing the starflower-gathering rates of blue and red sprites.
2. Write Down the Ratios: The rate for the blue sprite is $$10$$ starflowers to $$2$$ hours $$(10:2)$$, and for the red sprite it’s $$15$$ starflowers to $$3$$ hours $$(15:3)$$.
3. Simplify the Ratios: The blue sprite gathers at a rate of $$5$$ starflowers per hour $$(5:1)$$, and the red sprite also at $$5$$ starflowers per hour $$(5:1)$$.
4. Compare the Ratios: The rates are the same, so both sprites gather starflowers at the same rate!

And so, our mathematical adventure through the enchanted forest of comparing ratios comes to a close. Armed with the knowledge and skills we’ve gained, we’re ready to solve any word problem that comes our way! Until our next mathematical quest!

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