6:7 = 6/7 = 0.857

4:5 = 4/5 = 0.8

Solution: 6:7 > 4:5 (6:7 is greater than 4:5).

So to make this simple, ratios are basically fractions. When you want to compare ratios, you generally convert them into fraction form. If you are allowed, simplify the fraction into decimals to get the exact/approximate values. After that, you are able to compare easily.

I'm not sure what grade you are in but I'm sure that since this is basic math, you aren't able to use a calculator. It's obvious that using a calculator is a no brainer for this, and is obviously a boring method. It won't help you learn what you need to learn at this stage. But, there's two ways you can compare fractions easily without using a calculator and it'll get your brain going.

1) Long division (the division algorithm of two integers):

- Put the number of denominator on the outside and the number of the numerator on the inside. So for 6:7, the 7 goes on the outside and the 6 goes on the inside. For 4:5, the 5 goes on the outside and the 4 goes on the inside.

- Then you solve the long division for both ratios. Make sure you know your basic multiplication, put your zeros and decimals in the right spot, and bring down numbers as necessary to subtract.

- Lastly, compare the two. We normally use the symbol ">" for greater than and the symbol "<" for less than.

2) Pie chart/rectangle square method (only works for small numbers):

- Make sure your ratios are converted to fraction form. In this case, 6:7 = 6/7 and 4:5 = 4/5.

- Draw a circle or rectangle for each ratio.

- Then for each, draw triangles/squares that coordinate to the amount in the denominator (biggest number usually). So that is 7 and 5 respectively.

- For ratio 6:7, you want to shade 6 out of the 7 triangles/squares. For the ratio 4:5, you want to shade 4 out of the 5 triangles/squares.

- Now compare and figure which one is greater than the other. They're pretty close tbh, but 6/7 is the larger one in this case.

- You can then write 6:7 > 4:5 as your answer.

Note: The pie chart/rectangle square method isn't the most accurate for comparing since you can be easily fooled, but it's good for small numbers. This method isn't practical but it is a semi-fast way to compare. However, the long division method is much prefered because it's the most accurate and approximative you can get. The long division method will get your algorithmic thinking and mental mathematics skills on point when you need it later on. The long division method also requires cleanliness during the algorithm because you can easily make mistakes if you aren't careful. Otherwise, it's a good challenge.

Ratios will come up in basic probability as well. If you are able to master the tougher aspects of this and have some general algorithmic thinking, you'll blast through the probability unit. Probability is quite annoying and tedious if you don't have some common sense and know your basic mathematics skills that you are taught at your specific grade level; such as counting, guessing, ratios, fractions, mean/median/range, etc.

Mathematics is tough for a lot of people, and even I struggle immensely on subjects that I should grasp at my age.

Let me know if I made any mistakes.