Master Ratio and Proportion Word Problems - Math

Table of Contents

1. Introduction
2. Ratios and Proportions
   • Understanding Ratios
   • Converting Ratios to Fractions
3. Problem 1: Cats and Dogs on an Island
   • Setting up the Ratio
   • Simplifying the Fraction
4. Problem 2: Boys and Girls in a Class
   • Setting up the Proportion
   • Solving for the Unknown
5. Problem 3: Cakes Made in a Given Time
   • Setting up the Proportion
   • Finding the solution
6. Problem 4: Width of Large Rectangle
   • Setting up the Proportion
   • Calculating the Width
7. Problem 5: Number of Nickels in a Jar
   • Setting up the Ratios
   • Solving for the Unknown
8. Conclusion

Ratios and Proportions: Solving Problems Step by Step

Ratios and proportions are essential concepts in mathematics that can help us solve various problems involving relationships between different quantities. In this article, we will explore how to solve problems using ratios and proportions, providing step-by-step solutions to specific examples.

Problem 1: Cats and Dogs on an Island

Let's start with a problem that involves finding the ratio of cats to dogs on an island. Suppose there are 540 cats and 675 dogs. We can set up a fraction to determine this ratio. Since cats came first, we put their count on the top of the fraction, while dogs come Second, so we place their count on the bottom.

To simplify the fraction, we Notice that both 540 and 675 are divisible by 5 since their last digits are either zero or five. Dividing 540 by 5 gives us 108, and dividing 675 by 5 gives us 135. Further simplifying, we find that both 108 and 135 are divisible by 9. Dividing 108 by 9 yields 12, and dividing 135 by 9 gives us 15.

Hence, we can cancel out the common factor of three and conclude that the ratio of cats to dogs on the island is 4:5 or, expressed with a colon, 4 to 5.

Problem 2: Boys and Girls in a Class

In this problem, We Are given the ratio of boys to girls in a class as 8 to 7. If there are 40 boys, we need to determine the number of girls in the class.

We set up a proportion by writing two fractions separated by an equal sign. The ratio of boys to girls is 8 to 7, so if there are eight boys, there will be seven girls in the same class. Now, if there are 40 boys in the class, we can represent the number of girls, denoted by x, in proportion to the number of boys.

Cross-multiplying these fractions, we get 8 times x equals 40 times 7. Simplifying further, we calculate that 40 times 7 is 280. To isolate x, we divide both sides of the equation by 8. Therefore, the number of girls in the class is 35.

Conceptually, we can understand this problem by expanding the ratio. To go from 8 to 40 boys, we need to multiply by 5. Similarly, to keep the ratio the same, we must multiply 7 by 5, which gives us 35 girls. This method offers a mental shortcut to the answer.

Problem 3: Cakes Made in a Given Time

Imagine Karen can make 14 cakes in six hours, and we want to determine how many cakes she can make in 15 hours. To solve this problem, we set up a proportion with two fractions representing the number of cakes made and the corresponding time.

Since Karen can make 14 cakes in six hours, we set the first fraction as 14/6. To find the number of cakes she can make in 15 hours, we represent this unknown quantity as x.

Cross-multiplying the fractions, we get 6 times x equals 14 times 15. Simplifying further, we find that 14 times 15 is 210. Dividing both sides of the equation by 6, we discover that Karen can make 35 cakes in 15 hours.

As a quick mental calculation, we can note that the ratio of cakes to hours is initially 14 to 6. To find the number of cakes or x in 15 hours, we need to determine what number to multiply 6 by to reach 15. Dividing 15 by 6 gives us approximately 2.5. Therefore, we multiply 14 by 2.5, yielding the same result of 35 cakes.

Problem 4: Width of Large Rectangle

Suppose a small rectangle has a length of 9 inches and a width of 8 inches. The length of a larger rectangle is 24 inches. If the length and width of the two rectangles have the same ratio, we need to find the width of the larger rectangle.

To solve this problem, we set up a proportion using two fractions. The top part of the fractions represents the length, while the bottom part corresponds to the width. For the small rectangle, we have a length of 9 inches and a width of 8 inches. We denote the width of the larger rectangle as x.

Cross-multiplying these fractions, we find that 9x equals 8 times 24. To simplify the calculation, we can rewrite 9 as 3 times 3 and 24 as 8 times 3. Therefore, we have 3x on the left side and 8 times 8, which is 64, on the right side.

Next, we divide both sides of the equation by 3. However, 3 does not evenly divide into 64, so we leave our answer as 64/3. This fraction represents the exact width of the larger rectangle. Alternatively, we can express it as approximately 21.3 or 21.3 repeating in decimal form. Regardless, the units remain inches.

Problem 5: Number of Nickels in a Jar

Consider a jar containing a mixture of nickels, dimes, and quarters in the ratio of 3 to 4 to 7, respectively. If there are a total of 112 coins in the jar, our goal is to determine the number of nickels.

We set up four fractions to represent the ratios of nickels, dimes, quarters, and the total number of coins. The ratio is 3 to 4 to 7, so we have 3/14 on the top of the fractions. The bottom part represents the actual number of coins in the jar, which is 112.

To find the number of nickels, we set up the following equation: 3/n = 14/112. Cross-multiplying, we compute that 14n is equal to 3 times 112, which simplifies to 336.

By dividing both sides of the equation by 14, we find that n, the number of nickels, is 24. Thus, there are 24 nickels in the jar.

Additionally, we can observe the ratios for each Type of coin: 3 nickels, 4 dimes, and 7 quarters. To derive these values, we multiply each ratio by 8, as 8 times 3 is 24 and 8 times 7 is 56. The result confirms that the total number of coins in the jar is indeed 112.

Conclusion

Understanding ratios and proportions allows us to solve a variety of mathematical problems. By step-by-step analysis, we can determine the relationships between different quantities and find solutions efficiently. Whether it's calculating ratios of cats to dogs, boys to girls, cakes made in a given time, Dimensions of rectangles, or numbers of coins in a jar, ratios and proportions form the foundation of problem-solving in mathematics.