Ratios, Rates & Proportional Reasoning
Setting up and solving proportions, rates, ratios, and scale/similarity problems — includes map scales, similar figures, and unit rate problems
Generate Unlimited Practice Questions
Sign up for free and get 50 practice questions to start your prep.
Start Free PracticeLearn This Topic
Have you ever made the ultimate batch of slime? To get it perfectly stretchy, you might need 2 squirts of activator for every 3 cups of glue. Guess what? You just used a ratio! 🧪
A ratio is simply a way to compare two different things. It tells us how much of one thing there is compared to another. You can write a ratio with a colon like , with words like "2 to 3", or as a fraction like . Whether you are comparing chocolate chips to cookies, or dogs to cats at the pet store, ratios are everywhere! 🐶
A rate is a special kind of ratio that compares two completely different units. Think about the speedometer in a car—it measures "miles per hour." Or think about buying your favorite candy—the price might be "dollars per candy bar." When we figure out how much just ONE of something costs, or how far you travel in just ONE hour, that is called a unit rate.
On the ISEE test, you will see ratio and rate questions in both the Quantitative Reasoning and Mathematics Achievement sections. Sometimes they will ask you to compare two different unit rates in a Quantitative Comparison question (like which bag of chips is the better deal). Other times, they will give you a ratio of boys to girls and ask for the total number of students. The biggest secret to mastering ISEE ratio questions? Treat them just like fractions! If you can simplify a fraction, you can conquer any ratio problem the ISEE throws your way. 🚀
Practice Questions
5 practice questions for ISEE Middle Level
Show Solution
- Set up the proportion: . Cross-multiply: . Divide both sides by 3: . The baker needs 6 cups of flour.
Show Solution
- The scale is , which simplifies to 25 miles per inch. Multiply by the map distance: miles. Alternatively, set up the proportion and cross-multiply: , so miles.
Show Solution
- Set up the proportion . Cross-multiply: . Divide by 5: . There are 14 cashews.
Show Solution
- The unit rate is miles per gallon. At 10 gallons: miles. Alternatively, set up the proportion , so and .
Show Solution
- First, convert 15 tokens to coins: coins. Then convert 6 coins to gems: gems.
Tips & Strategies
- Always label your units when setting up a proportion! 🏷️ If you have on the left side, make sure you have on the right side.
- When a question gives you a ratio (like 3:5) and a TOTAL number, add the ratio numbers together (3 + 5 = 8) to find the 'total parts'. This is the magic key to unlocking the problem! 🔑
Common Mistakes
- Watch out for confusing 'part-to-part' ratios with 'part-to-whole' ratios! 🚨 If the ratio of cats to dogs is , the ratio of cats to TOTAL pets is , not .
- Don't forget that a 'unit rate' means the denominator is 1. If you know 4 apples cost $2.00, don't stop there! Find the cost of 1 apple () before comparing. 🍎
Frequently Asked Questions
Can I use a calculator on the ISEE to divide big numbers for unit rates?
Nope! Calculators aren't allowed on any part of the ISEE. 🧮 But don't worry, the test makers use friendly numbers that usually divide perfectly. Practice your times tables!
What if the Quantitative Comparison columns look exactly the same but are written differently?
This happens a lot! ⚖️ If Column A is and Column B is , they simplify to the exact same thing. In that case, you would choose (C) The columns are equal.
How do I know if a question is asking for a rate or a ratio?
A ratio usually compares the same type of things (like boys to girls). A rate compares two different units, often using the word 'per' (like miles 'per' hour or cost 'per' item). ⏱️
Is there a penalty if I guess wrong on a hard ratio question?
Never! 🎯 The ISEE does not penalize you for wrong answers. If you're stuck on a tricky proportion, eliminate the crazy answers and guess. Never leave a bubble blank!