TABE Level D Math: Numeration

There are three kinds of numeration problems on the TABE D.

The first kind is about place value. For example, the number 371 has the same digits as the number 317. How and why are they different?

The second kind is about comparing numbers. Which number is bigger?

The third kind is about fractional parts of a whole amount (like what part of your day is spent watching TV). 

Place Value
In order to understand place value, you need to know about exponents. An exponent is a way of showing a number multiplied by itself. For example, you can write 2 × 2 × 2 as 23. The raised number is called an exponent. The other number is called the base. In this example, 2 is the base, and 3 is the exponent. The exponent shows how many times the base is multiplied. Since 2 × 2 = 4, and 4 × 2 = 8, 2 × 2 × 2 = 8, or 23 = 8.

Exponents are sometimes called powers. For example, you read 23 as 2 to the power 3.
The number 10 is used as the base for writing numbers. Each place value is a power of 10. The units place stands alone. The tens place is 10 to the power 1 (10 = 101). The hundreds place is 10 to the power 2 (100 = 102). The thousands place is 10 to the power 3 (1000 = 103). And so forth.

You can break a number such as 2836 down into its place values:

Decimal numbers such as .253 can be broken down the same way, except the powers of ten are negative numbers. The place value just to the right of the decimal is tenths. The tenths place is 10 to the power −1(10−1). The hundredths place is 10 to the power −2(10−2). The thousandths place is 10 to the power −3(10−3).

You can break the number .253 breaks down into its place values:

Be careful to pay attention to the difference between tens and tenths, hundreds and hundredths, thousands and thousandths, and so forth.

The chart below shows place values to the right and left of the decimal point. Each column in this chart shows a place value. Place values get larger as you move from right to left.


Comparing Numbers
The words “larger than” and “smaller than” are used to compare numbers.

For example, 6 is larger than 5, and 2 is smaller than 10. The symbol abbreviation for “larger than” is >, and the symbol abbreviation for “smaller than” is <.

In math, means 6 is larger than 5, and means 2 is less than 10.

Notice that the symbol always points to the smaller number.

Comparing Fractions
If two fractions have the same denominator (bottom number), just compare the top number.

For example because the bottoms are the same and .

Suppose two fractions have different denominators, such as and

Find a common denominator (a number than both 3 and 4 divide evenly into).

Then change the two fractions to two fractions with the same denominator: and

Now compare and Since ,

Therefore,

Comparing Decimals
When you compare two decimals, look at one place value at a time, starting from the left. When you compare .3 and .4 you conclude that .4 is larger because 4 is larger than 3, and the two numbers have the same place value (tenths). If you compare .3 and .04 you would conclude that .3 is larger because you compare the numbers in the tenths place value (3 and the 0), and . If the digits are tied in a position, you move to the next position and compare. For example, when comparing .56 and .57 you conclude that .57 is larger because the 5s are tied and

Comparing Decimals to Fractions
If one number is a fraction and the other is a decimal, it’s generally easiest to change the fraction to a decimal. Then compare the two decimals. Use a calculator to change the fraction to a decimal. Divide the numerator by the denominator (numerator ÷ denominator).

For example, if you compare with .72, divide 3 by 4 on a calculator (3 ÷ 4) and get the answer .75.

Then compare .72 and .75, and conclude that .

 

Finding Part of a Whole
Remember that a proper fraction is one way of representing part of a whole.

For example, the fraction represents 5 slices of a pizza that has been cut into 8 slices.

Suppose you had a party and ordered 16 pizzas. If you only ate of the 16 pizzas, how many pizzas did you have left? To find of 16, multiply .

On a calculator, this would be 5 ÷ 8 × 16, which equals 10. Because 10 pizzas were eaten, 6 (16 − 10) were left.

To find a fractional part of a number, multiply the fraction by the number. Be careful to read the problem carefully to answer the right question.

Numbers (including whole numbers, fractions, and decimals) are pictured on a horizontal line. Because there is no largest or smallest number, the line extends forever in both directions.

Arrowheads at both ends indicate this.

A typical portion of a number line is shown below.

Zero is in the center. Positive numbers extend out to the right. Negative numbers extend out to the left.

The numbers get larger as you move from left to right anywhere on the number between two labeled numbers. Points on the line represent numbers.

The numbers −3 and 4 are labeled A and B. A. problem might ask you to locate a number that is between two labeled numbers.

For example, look at the problem below. Which point—A, B, C, or D—represents the number 4.3? There are 10 tick marks (the short vertical lines) from 4 to 5. Each of these is one-tenth (.1). The number 4.3 is the third tick mark to the right of 4. The answer is B.


Numeration Practice
Name the place value of the underlined digit in each number.

1.

2.

3.

4.

5.

6. Which number is eighty-three thousand, fifty nine? A. 80,359 B. 83,059 C. 803,059 D. 8,003,059

7. Which number is seven hundred five million, forty-one thousand, four hundred eight? A. 70,541,408 B. 700,541,408 C. 705,041,408 D. 7,005,541,408

8. Which of the following is another way to show the number 30,000 + 1000 + 600 + 70 + 9? A. 31,070 B. 31,670 C. 31,670 D. 31,679

9. Which of these is another way to write 2407? A. 2 × 102 + 4 × 101 + 7 B. 2 × 103 + 4 × 102 + 7 × 101 C. 2 × 103 + 4 × 102 + 7 D. 2 × 104 + 4 × 103 + 7

10. Which of these is another way to write 4,396,302? A. 4 × 105 + 3 × 104 + 9 × 103 + 6 × 102 + 3 × 101 + 2 B. 4 × 106 + 3 × 105 + 9 × 104 + 6 × 103 + 3 × 102 + 2 C. 4 × 106 + 3 × 105 + 9 × 104 + 6 × 103 + 3 × 102 + 2 × 101 D. 4 × 107 + 3 × 106 + 9 × 105 + 6 × 104 + 3 × 103 + 2

11. Which of these is another way to write .534? A. 5 × 10−2 + 3 × 10−1 + 4 B. 5 × 10−1 + 3 × 10−2 + 4 × 10−3 C. 5 × 102 + 3 × 101 + 4 D. 5 × 102 + 3 × 10−3 + 4 × 10−4

12. Which of these is another way to write .9021? A. 9 × 100 + 2 × 10−1 + 1 × 10−2 B. 9 × 10−1 + 2 × 10−2 + 1 × 10−3 C. 9 × 10−1 + 2 × 10−2 + 1 × 10−4 D. 9 × 10−1 + 2 × 10−3 + 1 × 10−4

13. Bill plans to walk 12 miles in a walk-a-thon.

What is of this distance? A. 4 miles B. 6 miles C. 8 miles D. 3 miles

14. Ladds Clothing is having a off sale on every item in the store. Crescent Clothing is taking off every item. Roberts is taking 35% off all items. Which store is reducing its prices by more? A. Ladds B. Crescent C. Roberts D. All equal

15. The Johnston family is planning a vacation. They will spend 3 days in New York City, 2 days in Niagara Falls, 1 day in Cooperstown, and 4 days camping in the Adirondack Mountains.

What fraction of their vacation will be spent in Niagara Falls?

A. B. C. D.

16. What number is represented by Point A on the number line? A. −6.8 B. −6.2 C. −5.8 D. −5.2

Answers: Numeration Practice

1. thousands

2. hundredths

3. hundred thousands

4. thousandths

5. tenths

6. b

7. c

8. d

9. c

10. b

11. b

12. d

13. a

14. b

15. b

16. d