Estimate and Convert Units Within One System

To estimate well, you need a picture of each unit in your mind. A benchmark is a familiar object whose size you already know. You compare other things to it.

Units measure different kinds of amounts:

• Length is how long, tall or far something is.
• Capacity is how much a container can hold.
• Weight is how heavy an object is. Customary ounces and pounds measure weight.
• Mass is how much matter an object has. Metric grams and kilograms measure mass.

The two tables below give benchmarks for some common units. Treat each benchmark as close to the unit's size. It helps you estimate. It does not replace measuring.

To estimate a measurement, ask two questions.

1. What kind of amount is it? Length, capacity, weight or mass. This tells you which units could work.
2. How many benchmarks would it take? Picture the benchmark repeated until it matches the object.

Try it. Does a bathtub hold about $40$ cups or about $40$ gallons? Both units measure capacity. A cup is about what a small school milk carton holds, and $40$ cartons would barely cover the bottom of the tub. A gallon is about what a large milk jug holds, and $40$ jugs would fill the tub. So about $40$ gallons is the sensible estimate.

A good unit gives a number that is easy to picture. Small units usually suit small things, and large units usually suit large things.

Now for converting. Picture a table edge that is exactly $3$ feet long. You can count $3$ feet along it, or you can count $36$ inches. The edge did not change. Only the unit you counted with changed.

A conversion fact tells how two units are related, such as $1$ foot $= 12$ inches. Read it as a unit rate, which you already know: $12$ inches per foot. That is why $3$ feet is $3 \times 12 = 36$ inches.

Notice the trade. An inch is smaller than a foot, so it takes more inches to cover the same length. A foot is larger, so it takes fewer feet. For any amount greater than zero, the smaller unit gives the larger number and the larger unit gives the smaller number.

Metric units are built on tens, so each conversion number in the next table is $10$, $100$ or $1{,}000$. The start of a unit's name, called a prefix, tells you its size.

• Kilo- means $1{,}000$. One kilometer is $1{,}000$ meters.
• Centi- means one hundredth. It takes $100$ centimeters to make $1$ meter.
• Milli- means one thousandth. It takes $1{,}000$ milliliters to make $1$ liter.

The rule does not change. Multiply by $k$ to reach a smaller unit. Divide by $k$ to reach a larger unit, which you can always do because $10$, $100$ and $1{,}000$ are not zero.

The arithmetic gets easier. Multiplying by $10$, $100$ or $1{,}000$ makes each digit worth that many times as much, so the digits shift one, two or three places to the left. Dividing shifts them the same number of places to the right.

Take the trail sign. A meter is smaller than a kilometer, so multiply: $2.5 \times 1{,}000 = 2{,}500$. The trail is $2{,}500$ meters long.

Here is the big idea. A measurement is a number and a unit. You can change the unit without changing the amount.

• Benchmarks help you choose a sensible unit and spot an answer that cannot be right.
• To convert inside one system, find the conversion number $k$, which is never zero. Multiply by $k$ to reach a smaller unit. Divide by $k$ to reach a larger unit.
• Check by converting back. An exact quotient times the divisor equals the dividend. A rounded quotient only comes close. A whole quotient with a remainder checks as the divisor times the whole quotient, plus the remainder.

Next, you will use ratio reasoning to convert units, and you will convert between the customary and metric systems.