Homeschool Math Blog

Math Mammoth Ratios, Proportions & Problem Solving is a worktext that concentrates, first of all, on two important concepts: ratios and proportions, and then on problem solving. It is meant for grades 6-7. This book has been now updated to include many new lessons that will ALSO be in the upcoming Math Mammoth grade 7-B. This means that you can use it to continue pre-algebra studies after finishing 7-A. See free samples and more info: http://www.mathmammoth.com/ratios_proportions_problem_solving.php For now, I've kept the download price at $5.00 though the book became quite a bit longer.

A few more free worksheet generators I have made recently. Again... there may be some minor 'bugs' in these, and if so, please let me know. Worksheets for the greatest common factor (GCF) and the least common multiple (LCM) Worksheets for basic ratio word problems Both of these are targeted especially for 6th grade.

Here's a word problem that someone sent me recently: The total number of girls in the combined junior and senior classes is equal to the number of boys in those two classes. If the senior class has 400 students and the junior class has 300 students, and if the ratio of boys to girls in the senior class is 5:3, what is the ratio of boys to girls in junior class? This problem gives a lot of information, and it sounds like it can be solved many different ways. But the first task is to notice what we are given and what we are asked . We are asked about a ratio .  We're given one ratio, and all kinds of totals. There are boys and girls , senior and junior classes. In other words, there are four groups: senior boys, senior girls, junior boys and junior boys. This sounds like it can be solved by setting up some equations and using algebra. To start, you could for example notice the two facts given about the senior class: The senior class has 400 students and the ratio of...

Simplify the ratio 186:403. The answer is 6:13. How do we get it? To simplify ratios (or fractions), we need to find COMMON FACTORS of the two numbers. So, one way to do it is to first find the GCF (Greatest Common Factor) of 186 and 403. Then divide 186 and 403 by it. Alternatively, just find ANY factor of 186 and 403, and divide both by it, to simplify the ratio somewhat, and to get started. Then repeat the process. Okay, 403 is not divisible by 2, 3, 4, 5, 6. This I know by divisibility tests. Maybe it's divisible by 7... need to try (calculator). No, it isn't. Maybe by 11? No. Maybe by 13? YES. My calculator helps. 403 = 13 x 31. I happen to know both of these are primes, so therefore 403 doesn't have any other factors. Then 186.... is it divisible by 13 or 31? By 13, no. By 31, YES!  186 = 31 x 6 So since 186 = 6 x 31 and 403 = 13 x 31, then the ratio 186:403 simplifies to 6:13. Clearly that's as far as we can get, as it's simplified to the lowest...

I guess it is time for some more problem solving, since someone sent this question in. Two numbers are in the ratio of 1:2. If 7 be added to both, their ratio changes to 3:5. What is the greater number? We can model the two original numbers with blocks. 1 block and 2 blocks makes the ratio to be 1:2. |-------| |-------|-------| Now add the same thing to both (the 7): 7 |-------|---| |-------|-------|---| 7 The way I just happened to draw these suggests that I could just split the original block in two, and the problem is solved: 7 |---|---|---| |---|---|---|---|---| 7 Here, each little block is 7. The original larger blocks are 14 each. So the original bigger number, which had two larger blocks, is 28, and the smaller number is 14. Check: Their ratio is 28:14 = 2:1. If you add 7 to both, you have 35 and 21, and their ratio is 35:21 = 5:3. Solving the same problem using algebra The two numbers in the ratio of 1:2 are x and 2x. Once 7...

Problem: If a:b = 1:3 and b:c = 3:4, find a:c. Two ratios are given, third is to be found. This is very very simple. The picture shows the two given ratios as blocks. We can see that a is one block and c is four blocks, so the ratio a:c is 1:4. You don't need an image for that, of course, since the original ratios are so easy. If a:b=1:3 and b:c=3:4, b being the same in both cases, we can write the ratio a:b:c as 1:3:4 right off. But what if the numbers weren't so friendly? What if it said this way: If a:b = 1:3 and b:c = 5:7, find a:c. This is solvable in various ways. I'll use equivalent ratios, in other words change the given ratios to equivalent ratios until we find ones where the b 's are the same. In the first ratio, 1:3, b is 3. In the other ratio, 5:7, it is 5. We can make those to be 15 by changing the ratios to equivalent ratios - which is done in an identical manner as changing fractions to equivalent fractions. 1:3 = 5:15 and 5:7 = 15:21. Now the ratio o...

Dave at MathNotations had an interesting ratio problem : In Virtual HS, the ratio of the number of juniors to seniors is 7:5. The ratio of (the number of) junior males to junior females is 3:2. The ratio of senior males to senior females is 4:3. What is the ratio of junior males to senior females? He asked if it can be solved using "Singapore" style bar model. I'm not sure if this is exactly how they'd do it, but this is how I'd do it... so here goes. After I made the diagrams, I soon saw that Dave's numbers are two awkward; the bar diagram drawing would get too messy because we'd need to divide it into too tiny parts to see anything. BUT... you probably know about the PROBLEM SOLVING STRATEGY called "solve an easier problem". My agenda is therefore: show how to solve a few related simple ratio problems using the bar diagram solve a variant of the original problem (with friendly numbers) solve the original problem. 1. Here's a bar diagram re...