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

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!