Solving Ratio Problems with Tables
Solving ratio problems with tables involves using a structured format to represent and manipulate the relationships between two quantities. Ratios compare these quantities, providing insights into their relative sizes and enabling practical applications in everyday scenarios, such as cooking, construction, and mapping. A ratio table serves as an organizational tool, allowing individuals to visualize and calculate equivalent ratios easily. For instance, given that 3 cups of peanuts weigh 12 ounces, a ratio table can be constructed to explore how changes in the amount of peanuts affect their weight. By performing operations within the table, one can derive additional values, such as determining that 18 cups would weigh 72 ounces, thereby illustrating the multiplicative relationship between the two quantities. This method not only simplifies problem-solving but also reinforces foundational mathematical concepts prevalent in higher-level studies, such as calculus and geometry. Ultimately, ratio tables are valuable in maintaining the consistency of ratios while answering various questions related to quantity comparisons.
On this Page
Subject Terms
Solving Ratio Problems with Tables
Ratios are used in countless everyday applications from reading maps to converting measurements in cooking or construction. A ratio is the comparison of two quantities that conveys their relative size. For example, the statements "one inch equals 25 miles" or "add 2 T for every 3 cups of water" describe ratios that can be expressed as 1:25 or 2:3, respectively. Ratios are used to build scale models of large objects, such as cars, buildings, and even cities.
Overview
Though there is no documented evidence of the first use of ratio tables, the Ahmes Papyrus dating back to 1700 B.C. suggests that the Egyptians were adept at using ratios of triangle side lengths in construction. Early Greek mathematics is also replete with ratio relationships found in geometry. For example, Thales of Miletus (640-546 BC) reportedly calculated pyramid heights by recognizing common ratios between the height of an object and the length of its shadow. Given the lack of precise calculating tools, it is likely that these early mathematicians used informal methods similar to those promoted in ratio tables.
Ratio tables are simple organizational tools to record mental strategies and computations. Solving ratio problems using tables requires the ability to reason multiplicatively as to how changes in one quantity affect the other quantity. These ideas are critical to the notion of function and are at the heart of fundamental mathematical concepts in calculus, geometry, and trigonometry.
Ratio tables are a way to represent the relationship between two quantities and organize computations to solve ratio problems. Consider the statement: 3 cups of peanuts weigh 12 ounces. A ratio table can be constructed to illustrate the relationship between cups of peanuts and weight. See Table 1. By performing successive operations down the table a series of equivalent ratios can be formed. For example, doubling both quantities we see that 6 cups of peanuts would weigh 24 ounces, 12 cups would weigh 24 ounces, and so on. We can also reason down the table by adding 12 to the right column every time we add 3 to the left. Finally, we can work backwards to determine that 1 cup of peanuts would weigh 4 ounces by dividing the first row by 3.
A ratio table provides an efficient way to reason up and down and answer additional questions such as, how much will 18 cups weigh? (72 ounces) Or, if I a buy a 5 lb. bag of peanuts, how many cups will I have? (20 cups)
As demonstrated in this example, there are many ways to solve problems using ratio tables. Regardless of the pathway, it is important that the same multiplicative relationship between quantities is always maintained. To confirm, we can look across the table above to see that the weight in ounces is always 4 times the number of cups—thus maintaining a ratio of 1 cup to 4 ounces.
Bibliography
Cajori, Florian. A History of Mathematics. Vol. 303. American Mathematical Soc., 1991.
Lamon, Susan J. Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers. New York: Routledge, 2012.
Lobato, Joanne, Amy Ellis, and Rose Mary Zbiek. Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics: Grades 6-8. Reston, VA: National Council of Teachers of Mathematics, 2010.
Middleton, James A., and Marja Van den Heuvel-Panhuizen. "The Ratio Table." Mathematics Teaching in the Middle School 1.4 (1995): 282-288.