Unit Rates Associated with Ratios of Fractions - Seventh Grade Mathematics

Unit rates provide a standardized benchmark for comparison, extending beyond whole numbers to evaluate complex ratios where quantities themselves are expressed as fractions.

Understanding Ratios, Rates, and Unit Rates

A ratio is a mathematical comparison of two quantities through division. When a ratio compares two quantities measured in different units of measurement, it is termed a rate. Familiar rates include speed measured in miles per hour, price measured in dollars per pound, or water flow measured in gallons per minute.

               DEFINITIONS AND STRUCTURAL FORMS
   Ratio:     Comparison of two quantities (e.g., 3 cups to 5 cups)
   Rate:      Comparison of different units (e.g., 150 miles in 3 hours)
   Unit Rate: Rate scaled to exactly 1 unit of denominator (e.g., 50 miles/1 hour)

A unit rate is a rate in which the second quantity, the denominator, is scaled to precisely one unit. In earlier grades, unit rates typically involved whole numbers: if a vehicle travels one hundred twenty miles in three hours, dividing one hundred twenty by three yields forty miles per one hour. In seventh grade, we encounter scenarios where both the numerator and denominator are fractional quantities.

Ratios of Fractions as Complex Fractions

When one or both quantities in a rate are fractions, the relationship forms a complex fraction. A complex fraction is a fraction whose numerator, denominator, or both contain fractions.

               STRUCTURE OF A COMPLEX FRACTION
                       Numerator:   a / b
                       Denominator: c / d
                       Full Form:   (a / b) / (c / d)

To determine the unit rate associated with a ratio of fractions, divide the numerator fraction by the denominator fraction. Recall that dividing by a fraction is algebraically equivalent to multiplying by its reciprocal.

               COMPUTING FRACTIONAL UNIT RATES
              (a / b) / (c / d) = (a / b) * (d / c) = (a * d) / (b * c)

Suppose a runner completes three-fourths of a mile in one-fifth of an hour. To find the runner's speed in miles per hour, divide three-fourths by one-fifth:

Speed = (3/4 mile) / (1/5 hour)
      = (3/4) * (5/1)
      = 15/4
      = 3 and 3/4 miles per hour (or 3.75 mph)

In one full hour, the runner covers three and three-fourths miles.

Visualizing Fractional Unit Rates with Double Number Lines

A double number line provides an intuitive visual representation of how two related quantities scale together toward one whole unit.

Distance (miles): 0       3/4      6/4      9/4     12/4    15/4 (3.75 mi)
                  +--------+--------+--------+--------+--------+
Time (hours):     0       1/5      2/5      3/5      4/5      1 (1 hour)

By observing each one-fifth increment of an hour paired with three-fourths of a mile, you can trace the proportional expansion across five equal intervals until reaching one complete hour.

Scaling Recipes and Mixing Concentrations

Fractional rates arise frequently in cooking and chemical solutions, where ingredients are combined in fractional measures.

Example: A baker uses 1/3 cup of honey for every 1/2 cup of almond flour.
Find the unit rate of honey per cup of almond flour.

Rate = (1/3 cup honey) / (1/2 cup almond flour)
     = (1/3) * (2/1)
     = 2/3 cup of honey per 1 cup of almond flour

This unit rate allows the baker to scale the recipe to any quantity of flour instantly by multiplying the flour measurement by two-thirds.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. An automated industrial pump drains two-fifths of a water storage tank in two-thirds of an hour. Calculate the unit rate of tank drainage per hour.

Exercise 2. A cyclist pedals five-sixths of a kilometer in one-quarter of a minute. What is the cyclist's speed in kilometers per minute? Convert this speed to kilometers per hour.

Exercise 3. A landscape gardener spreads three-eighths of a cubic yard of mulch over five-sixths of a garden bed. Find the unit rate of mulch application per square yard of the garden bed.

Exercise 4. A painter uses two-thirds of a gallon of paint to cover three-tenths of a wall. How many gallons of paint are needed to cover the entire wall?

Solutions

Solution to Exercise 1. Divide the fraction of the tank drained by the fraction of the hour elapsed: two-fifths divided by two-thirds. Invert the divisor and multiply: two-fifths times three-halves. The factor of two in the numerator and denominator cancels, leaving three-fifths. The pump drains three-fifths of the water storage tank per hour.

Solution to Exercise 2. First find the speed in kilometers per minute by dividing five-sixths by one-fourth: five-sixths times four over one. Multiply the numerators to get twenty and denominators to get six, which simplifies to ten-thirds, or three and one-third kilometers per minute. To convert this rate to kilometers per hour, multiply the per-minute rate by sixty (since there are sixty minutes in one hour): ten-thirds times sixty equals six hundred divided by three, which equals two hundred kilometers per hour.

Solution to Exercise 3. Divide the quantity of mulch by the fraction of the bed: three-eighths divided by five-sixths. Invert the divisor and multiply: three-eighths times six-fifths. Multiply numerators: three times six equals eighteen. Multiply denominators: eight times five equals forty. Simplify eighteen-fortieths by dividing both numbers by two, yielding nine-twentieths of a cubic yard of mulch per garden bed.

Solution to Exercise 4. To find the paint required for one whole wall, calculate the unit rate of gallons per wall: two-thirds divided by three-tenths. Multiply two-thirds by ten-thirds: two times ten equals twenty, and three times three equals nine. The resulting fraction is twenty-ninths, which equals two and two-ninths gallons of paint to cover the entire wall.