Ratios And Proportions: Practice Problems Explained

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Ratios and Proportions: Practice Problems Explained

Hey guys! Let's dive into some ratio and proportion problems. Ratios are used to compare two quantities, showing how much of one thing there is compared to another. A proportion, on the other hand, is a statement that two ratios are equal. Understanding these concepts is super useful in everyday life, from cooking to calculating discounts. So, let's get started and make math a little less intimidating.

1. Creating Ratios with Specific Values

1) Ratios Equal to 1.4

So, the question asks us to find two ratios that simplify to 1.4. Remember, a ratio can be expressed as a fraction. So, we need to find fractions that, when you divide the top number by the bottom number, you get 1.4. Here's how we can approach this:

  • Understanding the Goal: We want to create ratios a:b such that a/b = 1.4. This means a = 1.4b.
  • Method 1: Simple Multiplication
    • Let's start with a simple whole number for b. If we let b = 1, then a = 1.4 * 1 = 1.4. But we usually want whole numbers in our ratios. So, let's multiply both by 10 to get rid of the decimal. That gives us a = 14 and b = 10. So, our first ratio is 14:10. And yes, we can reduce this further by dividing both sides by 2 to 7:5.
  • Method 2: Choosing a Different Base
    • Let's try b = 5. Then a = 1.4 * 5 = 7. This gives us the ratio 7:5 straight away. Already simplified.
  • Method 3: Getting Creative
    • How about b = 20? Then a = 1.4 * 20 = 28. So, another ratio is 28:20. Again, we could simplify this (divide both sides by 4) down to 7:5.

So, a solution would be 14:10 and 7:5. We actually found that both of these reduce to the same simplified ratio. The key takeaway here is that there are infinite possibilities. You just need to ensure that when you divide the first number by the second, you get 1.4. Get creative with the numbers you choose!

2) Ratios Equal to 4/15

Now, let's find two ratios that simplify to 4/15. This is similar to the previous problem, but this time we are dealing with a fraction.

  • Understanding the Goal: We need ratios a:b such that a/b = 4/15. This means 15a = 4b.
  • Method 1: Direct Multiplication
    • The easiest way to get a ratio equal to 4/15 is to simply use the fraction itself! Thus, 4:15 is a valid ratio. This is our base ratio.
  • Method 2: Multiplying Both Sides
    • Multiply both parts of the ratio by the same number. For example, if we multiply both 4 and 15 by 2, we get 8 and 30. So, the ratio 8:30 is also equal to 4/15. Think of it like creating equivalent fractions.
  • Method 3: Larger Multipliers
    • Let's multiply both parts of the ratio by, say, 5. This gives us 4 * 5 = 20 and 15 * 5 = 75. So, the ratio 20:75 is another valid answer. This is still equal to 4/15.

So, one solution could be 4:15 and 8:30. Again, the key is to understand that you can multiply both sides of the ratio by any number (except zero) and still maintain the same proportion. Keep it simple, and you'll find these problems pretty straightforward!

2. Finding Ratios of Numbers

Now, let's find the ratios of different pairs of numbers. Remember, the ratio of a to b is simply a/b, simplified if possible.

1) 21 to 7

The ratio of 21 to 7 is 21/7. To simplify this fraction, we divide both the numerator and the denominator by their greatest common divisor, which is 7. So, 21/7 = (21 ÷ 7) / (7 ÷ 7) = 3/1. Therefore, the ratio is 3:1.

2) 350 to 50

The ratio of 350 to 50 is 350/50. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 50. So, 350/50 = (350 ÷ 50) / (50 ÷ 50) = 7/1. Therefore, the ratio is 7:1.

3) 15 to 9

The ratio of 15 to 9 is 15/9. The greatest common divisor of 15 and 9 is 3. So, we divide both by 3: 15/9 = (15 ÷ 3) / (9 ÷ 3) = 5/3. Thus, the ratio is 5:3.

4) 7.2 to 6.4

The ratio of 7.2 to 6.4 is 7.2/6.4. To get rid of the decimals, we can multiply both numbers by 10, which gives us 72/64. Now, we need to simplify this fraction. The greatest common divisor of 72 and 64 is 8. So, we divide both by 8: 72/64 = (72 ÷ 8) / (64 ÷ 8) = 9/8. Therefore, the ratio is 9:8.

5) 2/7 to 2/14

The ratio of 2/7 to 2/14 is (2/7) / (2/14). To divide fractions, we multiply by the reciprocal of the second fraction: (2/7) / (2/14) = (2/7) * (14/2). Now, we can simplify by canceling out the 2s: (1/7) * (14/1) = 14/7. Finally, we simplify 14/7 to get 2/1. So, the ratio is 2:1.

3. Finding Ratios of Quantities

Now let's deal with quantities that have units. When finding the ratio of quantities, it’s important that they have the same units. If they don't, you'll need to convert them first.

1) 20 kg to 12 kg

The ratio of 20 kg to 12 kg is 20 kg / 12 kg. Since the units are the same (kg), we can simply divide the numbers: 20/12. The greatest common divisor of 20 and 12 is 4. So, we divide both by 4: 20/12 = (20 ÷ 4) / (12 ÷ 4) = 5/3. Thus, the ratio is 5:3.

2) 4 cm to 12 cm

The ratio of 4 cm to 12 cm is 4 cm / 12 cm. Again, the units are the same (cm), so we divide the numbers: 4/12. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4. So, 4/12 = (4 ÷ 4) / (12 ÷ 4) = 1/3. Therefore, the ratio is 1:3.

And that's it! We've covered how to create ratios with specific values, find the ratios of numbers, and find the ratios of quantities. Remember, the key is to simplify the fractions as much as possible and to make sure your units are consistent when comparing quantities. Keep practicing, and you'll become a ratio master in no time! Hope this helps, and happy math-ing!