Algebra of Mixed Measures Algebra of Mixed Measures (Definition, Types and Examples) - BYJUS

# Algebra of Mixed Measures

We are familiar with quantities like length, mass, volume, and time. We are also familiar with the units that we use to measure these quantities. Sometimes we use more than one unit to measure a quantity. For example, we measure height in feet and inches. Mixed measures are quantities measured using a combination of units. Learn more about mixed measures, instances for which we use mixed measures, and how we perform operations on them....Read MoreRead Less

## What are mixed measures?

As we have learned, there are various quantities of measurements such as length, weight, capacity, and time. There are also multiple units to measure each of these quantities.

Length is the measure of distance from one point to another. It is measured in inches, yards, feet, or miles.

Weight is the measure of how heavy an object is. It is measured in ounces (oz), pounds (lb) or tons.

Time is a measure used to compare the start and end of an event or the duration that has elapsed. It is measured in seconds (s), minutes (min), hours (hr), days, weeks, months or even in years.

Capacity is a measure of the amount of space an object occupies. It is measured in pint, quart or gallon.

Mixed measures are a combination of these measurements such as hours, minutes and seconds, or yards, feet, and inches.

For example, Natalie runs about 5 miles and 30 yards in a marathon. Natalie ran for 10 minutes and 40 seconds in this event. Natalie also weighs 126 lb and 2 oz. Additionally, Natalie drank 1 gallon and 2 pints of water throughout the marathon. We can observe that the units of measure are written in mixed measures, when talking about Natalie’s experience at the marathon!

## Conversion of units

While writing measures in smaller units, we multiply the larger one to arrive at the smaller unit. For example, if we have to convert from minutes to seconds, which is a smaller unit, we will multiply.

The conversion relationships between various units of length, weight, time and capacity can be used for this.

## How to use regrouping to write mixed measures?

Problems that involve regrouping for mixed measures can be solved by first solving the values with the smaller units. If in an addition operation the resultant sum corresponding to the smaller unit is large enough, then, we regroup it with the larger unit values.

Similarly, for subtraction operation, if the minuend is smaller than the subtrahend corresponding to smaller units, then, we regroup 1 unit from the larger unit value, and combine it with the smaller unit values.

## Solved Examples

1. Add 6 feet 3 inches and 3 feet 4 inches.

6 ft  3 in

+ 3 ft  4 in

$$\text{—————–}$$

We will add the inches first, starting from the right.

6 ft  3 in

+ 3 ft  4 in

$$\text{—————–}$$

7 in

Now, we will add the feet.

6 ft  3 in

+ 3 ft  4 in

$$\text{—————–}$$

9 ft 7 in

The sum is 9 feet 7 inches.

2. Subtract 3 hours 23 minutes from 6 hours 12 minutes.

6 h  12 min

–  3 h  23 min

First, we will begin subtraction with the minutes on the right side. Since 23 is greater than 12, we will regroup 1 hour as 60 minutes and combine that with 12 min to get 72 min.

$$\begin{array}{ccc} &^{5}\not{6}\text{ h}&^{72}\not{12}\text{ min}\\ -&~~~\text{3 h}&~~~~~\text{23 min} \end{array}\\ \text{———————————}\\ \begin{array}{ccc} &&~~~~~~~~~~~~~~~~~~~\text{49 min} \end{array}$$

Now, we will subtract the hours.

$$\begin{array}{ccc} &^{5}\not{6}\text{ h}&^{72}\not{12}\text{ min}\\ -&~~~\text{3 h}&~~~~~\text{23 min} \end{array}\\ \text{—————————–}\\ \begin{array}{ccc} &~~~~~~~~~~\text{2h}&~~~~~~\text{49 min} \end{array}$$

Thus, the difference is 2 h 49 min.

3. Melissa’s height was 4 feet 3 inches and she weighed 40 pounds 8 ounces. 2 years later, Melissa now has a height of 5 feet 5 inches and weighs 50 pounds 12 ounces. What is the height and weight gained by Melissa in these 2 years?

Answer: In order to find the height and weight gain by Melissa, we will have to find the difference between the mixed measurements given.

Let us first find the difference between the height. We will align the addends as per the units and subtract them.

5 ft   5 in

– 4 ft   3 in

$$\text{—————–}$$

We will subtract the inches first and then move to the feet, from right to left.

5 ft   5 in

– 4 ft   3 in

$$\text{—————–}$$

1 ft   2 in

The height Melissa has gained is 1 ft 2 in.

Now, we will find the weight gained by subtracting the weights. Here, we do not need to regroup while subtracting, as 12 ounces is greater than 8 ounces. We will begin from the right and move toward the left.

50 pounds 12 ounces

– 40 pounds  8 ounces

$$\text{——–————————}$$

10 pounds  4 ounces

Hence, Melissa gained 10 pounds 4 ounces.

4. Lemonade is prepared with 4 quarts 2 pints of lemon juice and 2 quarts 2 pints of water. How much lemonade is made with the given quantities of lemonade and water?

4 quarts  2 pints

+ 2 quarts  2 pints

$$\text{————————}$$

6 quarts  4 pints

6 quarts and 4 pints of lemonade is prepared.

5. The following table shows the timeline of the diet maintained by Anne and Jenny.

 Anne 45 days, 4 hours, 30 minutes Jenny 40 days, 5 hours, 30 minutes

In comparison, who has maintained a longer duration of the diet, and by how much longer?

From the above given table, we can see that Anne has maintained her diet for more days than Jenny.

Now, to know how much longer, we have to find the difference between the durations of the diet.

45 days  4 hours  30 minutes

– 40 days  5 hours  20 minutes

$$\text{—–————————————}$$

We will begin the subtraction from minutes first, then move to hours and then to days.

45 days  4 hours  30 minutes

– 40 days  5 hours  20 minutes

$$\text{——–————————————}$$

10 minutes

Since 5 is greater than 4, we will regroup 1 day as 24 hours and combine that with 4 hours to get 28 hours.

$$\begin{array} {cccc} &^{44}\not{44}\text{ days}&^{28}\not{4}\text{ hours}&\text{30 minutes}\\ -&^{}~~~~\text{40 days}&~~~~\text{5 hours}&~\text{20 minutes}\\ \end{array}\\\text{ —————————————————-}\\ \begin{array}{ccccc} &~~~~~~~~~~~~~\text{4 days}&~~~\text{23 hours}~~~\text{10 minutes} \end{array}$$

The difference in the duration of the diet maintained by Anne and Jenny is 4 days 23 hours and 10 minutes.