# Teaching Ratios & Unit Rates In Math

Financial ratios are relationships determined from a company’s financial information and used for comparison purposes. Examples include such often referred to measures as return on investment , return on assets , and debt-to-equity, to name just three.

How much would 10 pens cost? ” To solve the problem, find the unit price of the pens, then multiply by 10. In a continued proportion of four quantities, the first and the second proportional are . When four abstract numbers are in proportion, the product of the extremes is equal to the product of the means. We basically have to express the ratios as fractions and multiply both the fractions by the product of the two consequents.

## Developing The Concept: Rates

You could also find the unit rate by dividing the first term of the ratio by the second term. The heart of middle school mathematics, and a key part of algebra readiness, is understanding ratios and rates. The overview and lessons below are tools to prepare students, usually in Grades 6 and up, who are ready to learn about these concepts. The lessons below will typically cover two days of instruction. When the values of two ratios are equal, they are said to be in ‘proportion’ and one is called proportional to the other. Since a ratio is expressed as a fraction, its value is not changed when both its antecedent and consequent are multiplied or divided by the same number except 0. Antecedent and Consequent are similar quantities and in determining the ratio, they are to be expressed in the same unit.

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If three quantities be in continued proportion, the product of the first and the third is equal to the square of the second. Four quantities are said to be in proportion or are proportional when the ratio of the first to the second is equal to the ratio of the third to the fourth. In ratio of greater inequality, the antecedent is greater than the consequent.

## Leverage Ratios

These ratios are the result of dividing one account balance or financial measurement with another. Usually these measurements or account balances are found on one of the company’s financial statements—balance sheet, income statement, cashflow statement, and/or statement of changes in owner’s equity. Financial ratios can provide small business owners and managers with a valuable tool with which to measure their progress against predetermined internal goals, a certain competitor, or the overall industry. In addition, tracking various ratios over time is a powerful means of identifying trends in their early stages. Ratios are also used by bankers, investors, and business analysts to assess a company’s financial status.

• ” To solve the problem, find the unit price of the pens, then multiply by 10.
• You could also solve this problem by first finding the unit rate and multiplying it by 12.
• The ratio of two quantities of different kinds is not possible.
• In a continued proportion of four quantities, the first and the second proportional are .
• If so, point out that speed means calculating how fast something is going by comparing distance to time, such as miles to hours.
• Often, a small business’s ability to obtain debt or equity financing will depend on the company’s financial ratios.

Financial ratios can be an important tool for small business owners and managers to measure their progress toward reaching company goals, as well as toward competing with larger companies. Ratio analysis, when performed regularly over time, can also help small businesses recognize and adapt to trends affecting their operations. Often, a small business’s ability to obtain debt or equity financing will depend on the company’s financial ratios. Profitability ratios provide information about management’s performance in using the resources of the small business. Many entrepreneurs decide to start their own businesses in order to earn a better return on their money than would be available through a bank or other low-risk investments. However, it is important to note that many factors can influence profitability ratios, including changes in price, volume, or expenses, as well as the purchase of assets or the borrowing of money. Some specific profitability ratios follow, along with the means of calculating them and their meaning to a small business owner or manager.

## Proportion

As a unit rate example, you can show that the unit rate of 120 students for every 3 buses is 40 students per bus. Now that students know how to find a unit rate, they will learn how to find an equivalent ratio using unit rates. Finding equivalent ratios uses the same thought process as finding equivalent fractions. Collection period 365/Accounts Receivable Turnover—measures the average number of days the company’s receivables are outstanding, between the date of credit sale and collection of cash. It is important to keep in mind that financial ratios are time sensitive; they can only present a picture of the business at the time that the underlying figures were prepared. For example, a retailer calculating ratios before and after the Christmas season would get very different results. Ratios are calculated by dividing one number by another, total sales divided by number of employees, for example.

If a ratio is equal to the reciprocal of the other, then either of them is in ‘inverse proportion’ of the other. Conversion into same denominator, etc. can also be applied in the case of ratios. The ratio is an abstract number, it has no unit. The ratio of two quantities of different kinds is not possible. Looking for more free math lessons and activities for elementary school students? Be sure to explore our Free Teaching Resources hub. Compare the different ideas students offer.

Three quantities of the same kind are said to be in ‘Continued Proportion ‘when the ratio of the first to the second is equal to the ratio of the second to the third. The second quantity is called mean proportional between the first and the third and the third quantity is called third proportional to the first and second. Thus are in continued proportion for . Here, is the mean proportion between and and is the third proportional to and . There may be continued proportion in case of more than three quantities of the same kind. In general, for any odd number of quantities of the same kind, continued proportion will exist among the quantities. Accounts receivable turnover Net Sales/Average Accounts Receivable—gives a measure of how quickly credit sales are turned into cash.

## Financial Ratios

Have students independently calculate the unit price and answer which store has the better deal. Compare how different students did the calculations, and have students discuss similarities and differences among the models they used and solutions they found. Allow for varying answers such as “the second store had the better deal for me because I would only want two carrots anyway.” If time permits, have students solve the following problem as well. Let’s take a look at a problem that involves unit price. “A sign in a store says 3 Pens for \$2.70. This sort of multiplication is known as ‘Cross Multiplication’. In the ratio of lesser inequality, the antecedent is smaller than the consequent. You could also solve this problem by first finding the unit rate and multiplying it by 12.

## What Is A Unit Rate In Math?

Inventory to assets ratio Inventory/Total Assets—shows the portion of assets tied up in inventory. Generally, a lower ratio is considered better. In the ratio of equality, the antecedent is equal to the consequent. The ratio between two quantities of the same kind is called ‘simple ratio‘. For example, the inverse ratio of is and the inverse ratio of is . Perhaps the best way for small business owners to use financial ratios is to conduct a formal ratio analysis on a regular basis. The raw data used to compute the ratios should be recorded on a special form monthly. Then the relevant ratios should be computed, reviewed, and saved for future comparisons. Determining which ratios to compute depends on the type of business, the age of the business, the point in the business cycle, and any specific information sought. For example, if a small business depends on a large number of fixed assets, ratios that measure how efficiently these assets are being used may be the most significant.

Rates are used by people every day, such as when they work 40 hours per week or earn interest every year at a bank. When rates are expressed as a quantity of 1, such as 2 feet per second or 5 miles per hour , they can be defined as unit rates. You can write any rate as a unit rate by reducing the fraction so it has a 1 as the denominator or second term.

Alternatively, the reciprocal of this ratio indicates the portion of a year’s credit sales that are outstanding at a particular point in time. Virtually any financial statistics can be compared using a ratio. In reality, however, small business owners and managers only need to be concerned with a small set of ratios in order to identify where improvements are needed. Rate and unit rate are used to solve many real-world problems. Look at the following problem.

Evidently, inverse ratio or reciprocal of a ratio of greater inequality is a ratio of lesser inequality. Despite all the positive uses of financial ratios, however, small business managers are still encouraged to know the limitations of ratios and approach ratio analysis with a degree of caution. Ratios alone do not make give one all the information necessary for decision making. But decisions made without a look at financial ratios, the decision is being made without all the available data.

• It is important to keep in mind that financial ratios are time sensitive; they can only present a picture of the business at the time that the underlying figures were prepared.
• You could also find the unit rate by dividing the first term of the ratio by the second term.
• If three quantities be in continued proportion, the product of the first and the third is equal to the square of the second.
• Inventory to assets ratio Inventory/Total Assets—shows the portion of assets tied up in inventory.
• Antecedent and Consequent are similar quantities and in determining the ratio, they are to be expressed in the same unit.
• Three quantities of the same kind are said to be in ‘Continued Proportion ‘when the ratio of the first to the second is equal to the ratio of the second to the third.

The reciprocal of is and the reciprocal of is . Since the quotient of two quantities of the same kind is an abstract number, the ratio involving two quantities of the same kind is abstract. Have students try to figure it out individually. Compare student solutions, and discuss why Ebony runs 1.5 laps in one minute. Have students think of other examples of rates.

## Rates In The Real World

Ratios enable business owners to examine the relationships between items and measure that relationship. Ratios are aids to judgment and cannot take the place of experience. But experience with reading ratios and tracking them over time will make any manager a better manager. Ratios can help to pinpoint areas that need attention before the looming problem within the area is easily visible. Students may say that a rate is a ratio in which the quantities being compared use different units such as dollars and ounces or miles and hours. Students may use common English synonyms for rate such as speed. If so, point out that speed means calculating how fast something is going by comparing distance to time, such as miles to hours. If necessary, explain what a rate is. Although they may seem intimidating at first glance, all of the aforementioned financial ratios can be derived by simply comparing numbers that appear on a small busi-ness’s income statement and balance sheet. Small business owners would be well-served by familiarizing themselves with ratios and their uses as a tracking device for anticipating changes in operations.

Some students might recognize that if they divided 30 by 1.5, they would get 20 minutes. Discuss this strategy with the class. We know that Ebony can run 18 laps in 12 minutes. We also know she is going to run at that same rate for 30 laps.

## Teaching Ratios And Unit Rates In Math

In addition to common real-world examples, encourage silly or unusual rates such as hip-hop artists per zip code or diamond collars per chihuahua. Finding the cost of one unit enables you to find the cost of any number of units. When you find equal ratios, it is important to remember that if you multiply or divide one term of a ratio by a number, then you need to multiply or divide the other term by that same number. Removing the units makes the calculation easier to see. However, it is important to remember the units when interpreting the new ratio.

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“Tonya works 60 hours every 3 weeks. At that rate, how many hours will she work in 12 weeks? ” The problem tells you that Tonya works at the rate of 60 hours every 3 weeks. To find the number of hours she will work in 12 weeks, write a ratio equal to 60/3 that has a second term of 12. When a price is expressed as a quantity of 1, such as \$25 per ticket or \$0.89 per can, it is called a unit price. If you have a non-unit price, such as \$5.50 for 5 pounds of potatoes, and want to find the unit price, divide the terms of the ratio. Ratio of greater inequality and ratio of lesser inequality are called ‘ratios of inequality’.

In determining the ratio, the quantities are to be expressed in the same unit. Students will probably not know what a unit rate is, so provide them with the following explanation to explain rate vs. unit rate. The product of three continued proportional numbers is . Find the mean proportional.