Ratio and Proportion Word Problems - Math

The first problem asks for the ratio of cats to dogs on an island, given 540 cats and 675 dogs. The solution involves setting up a fraction (cats/dogs) and then simplifying it by dividing both numbers by common factors (5, then 9, then 3) to arrive at a simplified ratio of 4 to 5.

This problem presents a ratio of boys to girls (8 to 7) and states there are 40 boys. The goal is to find the number of girls. A proportion is set up (8/7 = 40/x), and cross-multiplication is used to solve for x. An alternative, quicker method involving a multiplier is also shown, where 8 multiplied by 5 equals 40, so 7 multiplied by 5 gives 35 girls.

Karen can make 14 cakes in 6 hours, and the problem asks how many cakes she can make in 15 hours. A proportion is set up as 14 cakes / 6 hours = x cakes / 15 hours. Cross-multiplication leads to 6x = 14 * 15, which solves for x = 35 cakes. A faster method finds the multiplier between 6 and 15 (2.5), and then applies it to 14 to get 35.

Given a small rectangle with length 9 inches and width 8 inches. A large rectangle has a length of 24 inches, and the length to width ratio is the same for both. The task is to find the width of the large rectangle. The proportion 9/8 = 24/x is set up. Cross-multiplication yields 9x = 8 * 24. Simplifying before multiplying by breaking down numbers helps in finding x = 64/3 inches or approximately 21.3 repeating inches.

This multi-step problem involves the ratio of nickels, dimes, and quarters (3:4:7) in a jar containing a total of 112 coins. The goal is to find the number of nickels. By adding the ratio parts (3+4+7=14), a combined ratio with the total is formed (nickels/total = 3/14). This is then set equal to the actual numbers (x nickels / 112 total coins) and solved via cross-multiplication for x. The method for finding the number of dimes and quarters based on the same multiplier is also demonstrated.