Ratios, Rates & Proportional Reasoning
Setting up and solving proportions, rates, ratios, and scale/similarity problems — includes map scales, similar figures, and unit rate problems
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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
3 practice questions for ISEE Upper Level
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- Let be the amount of cement needed. Set up a proportion relating the ratio of cement to sand: . To solve for , you can cross-multiply to get , and then divide by 9 to get . Alternatively, notice that the amount of sand (36) is 4 times the ratio part (9). Therefore, the amount of cement must be 4 times its ratio part: . The contractor needs 16 pounds of cement.
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- Set up a proportion comparing the map distance to the actual distance: , where is the actual distance in miles. Cross-multiply to get . To isolate , multiply both sides by the reciprocal, . This gives . The actual distance between the two towns is 100 miles.
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- Set up a proportion relating the height of the building to its width. The ratio of the actual height to the actual width is . Simplify this fraction by dividing the numerator and denominator by 30 to get . Let be the width of the model. Set the simplified ratio equal to the model's ratio of height to width: . Cross-multiply to get . Divide both sides by 15 to find . The width of the model is 8 inches.
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!