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 tried to make the ultimate batch of slime? 🧪 If you just throw in random amounts of glue and activator, you might end up with a sticky mess! To get that perfect, stretchy masterpiece, you need the right ratio. A ratio is just a mathematical way of comparing two things. For example, your slime recipe might need 2 spoons of glue for every 1 spoon of activator. We write that as a fraction: .
Rates are a special kind of ratio that compare two completely different units. Think about playing your favorite racing video game! 🏎️ When you zoom past the finish line, your speed is measured in "miles per hour." That's a rate! It tells you exactly how many miles you traveled in one hour of time.
On the SSAT, you'll see a lot of word problems that ask you to figure out ratios and rates. The test makers might ask you how fast a runner is going, or how to divide up a giant pile of pizza slices fairly among your friends. 🍕 Don't worry, you already use ratios and rates in real life all the time! The secret to beating these questions is simply keeping your units organized. If you treat these math problems like a fun puzzle or a secret recipe, you'll be a ratio master in no time!
Practice Questions
4 practice questions for SSAT Upper Level
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- First, find the total number of marbles in the jar by adding the number of red, blue, and green marbles: 14 + 10 + 6 = 30. The ratio of the number of red marbles to the total number of marbles is . To simplify this fraction, divide both the numerator and the denominator by their greatest common divisor, which is 2. This gives .
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- Let Maya's initial amount of money be and Julian's initial amount be . During the summer, Maya spent $20, so her new total is . Julian deposited $10, so his new total is . The new ratio is 1 : 1, which means they ended up with the same amount of money. Set their new totals equal to each other: . Subtract from both sides to get . Add 20 to both sides to get . Divide by 2 to find . The question asks for Maya's initial amount of money, which is . Substitute 15 for to get .
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- The initial ratio of water to vinegar is 3.5 to 0.75. To eliminate the decimals, multiply both numbers by 100, which gives 350 to 75. Next, simplify the ratio by dividing both numbers by their greatest common factor, which is 25. and . The simplified ratio of water to vinegar is 14:3.
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- To combine the two ratios, you need to find a common number for the cats, since cats appear in both ratios. The ratio of dogs to cats is 4:5, and the ratio of cats to rabbits is 3:2. The least common multiple of the cat portions (5 and 3) is 15. Multiply both parts of the dogs-to-cats ratio by 3 to get 12:15. Multiply both parts of the cats-to-rabbits ratio by 5 to get 15:10. Now you can combine them into a single ratio of dogs to cats to rabbits, which is 12:15:10. The ratio of dogs to rabbits is 12:10. Dividing both numbers by 2 simplifies this ratio to 6:5.
Tips & Strategies
- Always double-check your units! 📏 If a question gives you minutes but asks for 'miles per hour', you must convert the minutes to hours first.
- Use the 'Magic Triangle' for Distance, Rate, and Time. If you know two of them, you can find the third! .
- Think of ratios like baking a cake. If you double the flour on the top of your fraction, you must double the sugar on the bottom! Whatever you multiply the top by, multiply the bottom by the exact same number.
Common Mistakes
- Watch out for mixing up the order of a ratio! If the SSAT asks for the ratio of cats to dogs, make sure the number of cats goes on top: . Order matters!
- Don't forget that half an hour is NOT 50 minutes! It's 30 minutes, or of an hour. Time can be a tricky trap.
Frequently Asked Questions
What exactly is a unit rate?
A unit rate is just a ratio where the bottom number is 1. For example, '50 miles per 1 hour' or '3 cookies per 1 student' are unit rates!
How often do ratio questions show up on the SSAT?
You will definitely see them! They are very popular in the math section, especially as word problems involving speed, prices, or recipes.
What if I forget the formula for speed during the test?
Just picture a car's speedometer! It says 'mph' which stands for 'miles per hour'. That literally translates to 'miles divided by hours', giving you the formula .
Can I simplify ratios just like regular fractions?
Yes! Ratios act exactly like fractions. If you have a ratio of , you can divide the top and bottom by 4 to simplify it to .