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
5 practice questions for SSAT Middle Level
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- Set up a proportion using the ratio of dogs to cats: . Cross-multiply to solve for : , which simplifies to . Dividing both sides by 3 gives dogs.
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- First, find the unit rate for each exchange. Since 3 rubies = 12 sapphires, dividing by 3 shows that 1 ruby = 4 sapphires. Since 5 sapphires = 15 emeralds, dividing by 5 shows that 1 sapphire = 3 emeralds. If you have 4 rubies, they can be exchanged for sapphires. Those 16 sapphires can then be exchanged for emeralds.
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- Since the pump moves 4,200 gallons in a full hour, multiply the hourly rate by the fraction of the hour to find the amount moved: . The pump can move 700 gallons in hour.
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- The ratio of boys to girls is 4:5, which means for every group of 4 boys, there are 5 girls. This makes a total of students per group. Therefore, the total number of students in the club must be a multiple of 9. Looking at the choices, only 36 is a multiple of 9.
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- For every group of bulbs planted, there are 7 tulips and 4 daffodils. The total number of bulbs in one group is . The fraction of the bulbs that are tulips is the number of tulips divided by the total number of bulbs, which is .
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 .