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Section 1-1: Whole Numbers, Decimals, and the Place-Value System Learning Outcome 1 Identify the place value of each of the digits in the number 4,735: 4 thousands, 7 hundreds, 3 tens, 5 ones. Learning Outcome 2 The number 451,375 is written in standard form. Write 2,853 in words: two thousand, eight hundred fifty-three. Write 2,042 in expanded notation. 2 x 1,000 + 0 x 100 + 4 x 10 + 2 x 1 Multiply each digit by its place value and add the products. Learning Outcome 3 Write two inequalities comparing 4 and 7. 4<7 Using the less-than symbol, place the smaller number on the left. 7>4 Using the greater than symbol, place the larger number on the left. Learning Outcome 4 In 6.28, 6 is in the ones place, 2 is in the tenths place, and 8 is in the hundredths place. Learning Outcome 5 Write 57.035 in words: fifty-seven and thirty-five thousandths. Learning Outcome 6 Write the numerator as a decimal with as many decimal places as the denominator has zeros. Express 125 as a decimal: 0.0125. (The number has 4 decimal places because the 10, 000 denominator has 4 zeros.) Learning Outcome 7 Which decimal is larger, 0.45 or 0.445? Both numbers have the same digit , 4, in the tenths place. 0.45 is larger because it has a 5 in the hundredths place, while 0.445 has a 4 in that place. Learning Outcome 8 Round 4,734 to the nearest tens place: 3 is in the tens place; 4 is the digit to the right; 4 is less than 5, so round down. The rounded value is 4,730. Round 6.837 to the nearest hundredth: 3 is in the hundredths place; 7 is the digit to the right; 7 is 5 or more, so round up by raising the 3 to a 4 and dropping the 7. The rounded value is 6.84. Learning Outcome 9 Round 3,682 to a number with one nonzero digit: 3 is the first nonzero digit on the left; 6 is the digit to the right; 6 is 5 or more, so round up. The rounded value is 4,000. Round 0.0683 to a number with one nonzero digit: 6 is the first nonzero digit on the left; 8 is 5 or more, so round up by changing the 6 to a 7 and dropping the remaining digits on the right. The rounded value is 0.07. Section 1-2: Adding Whole Numbers and Decimal Numbers Learning Outcome 1 4+2=2+4 6=6 Numbers may be added in any order. (4 + 3) + 6 = 4 + (3 + 6) 7+6=4+9 13 = 13 Grouping does not change the sum. 0+5=5+0 5=5 Zero added to a number does not change the number. 4 + 5 = 9. Write 9 in the ones place. 2 + y = 6. Write 6 in the tens place. 7 + 7 = 14. Write 4 in the hundreds place and carry the 1 to next column. 5 + 1 carried = 6. Write 6 in the thousands place. 5,724 + 745 6,469 Learning Outcome 2 Add: 33.25 + 4.5 +0.123 a 33.25 4.5 + 0.123 37.873 Align decimals in each addend in a vertical line. Add each column starting at right. Keep decimal in answer in same vertical line as in addends. Add the columns and carry as needed. Learning Outcome 3 Estimate the sum by rounding to the nearest hundred: 372 + 645. 372 rounds up to 400. 645 rounds down to 600. 400 + 600 = 1,000. Check by adding the numbers a second time. Exact sum: 1,017. Add: 46.7 + 3,826 + 4,573 Calculator steps: 46[.]7 [+] 3826 [+] 4573 [=] Equal key may also be labeled as [ENTER] or [EXE] Sum = 8445.7. Section 1-3: Subtracting Whole Numbers and Decimals Learning Outcome 1 If 6 + 4 = 10, then 10 – 6 = 4 or 10 – 4 = 6. Addition and subtraction are inverse operations. 6–3=3 Subtract the digits in the ones column. To subtract 3 – 4 in the tens column, borrow 1 from the 5,436 4 in the hundreds column and make the 3 a 13; then 13 – 4 = 9. – 243 Remember in the hundreds column the borrowed 1makes the 4 5,193 a 3, so 3 – 2 = 1. Learning Outcome 2 Subtract: 68.029 – 6. 4276. 68.029 – 6.4276 61.6014 Align the decimals in a vertical line. In the thousandths column, borrow 1 from the 9 to make understood 0 in the ten-thousandth column a 10; then 10 – 6 = 4. In the ones column, borrow 1 from 8 to make the 0 in the tenths column 10; then 10 – 4 = 6. Remember, in the ones column 8 is now 7, 7 – 6 = 1. Learning Outcome 3 Estimate the difference by rounding to the nearest thousand: 4,752 – 2,641. 5,000 – 3,000 = 2,000. Exact difference: 4,752 – 2,641 = 2,111. Check: 2,111 + 2,641 = 4,752. To check add the difference and the subtractand. Section 1-4: Multiplying Whole Numbers and Decimals Learning Outcome 1 3×5=5×3 15 = 15 The order of the numbers multiplied does not matter. ( 4 × 2) × 3 = 4 × (2 × 3 ) 8 × 3=4 × 6 Grouping of numbers multiplied does not matter. 24 = 24 6 × 0=0 × 6 0=0 Multiplying by zero always gives zero as a product. 125 × 21 125 250 2, 625 Multiply multiplicand by ones digit of multiplier; put partial product so ones digits are in line. Multiply by tens digit of multiplier; place partial product so tens digits are in line. Add the partial products Learning Outcome 2 Multiply: 5.75 x 0.25 5.75 x 0.25 2875 1150 000 Multiply by each digit in the multiplier. Since there are four decimal places total in multiplicand and multiplier, make four decimal places in product. 1.4375 Learning Outcome 3 4(5 + 2) = 4(5) + 4(2) 4(7) = 20 + 8 28 = 28 The numbers in parentheses may be added (or subtracted) and then multiplied by number outside parentheses. Or each number in parentheses may be multiplied by number outside parentheses and then added (or subtracted). Learning Outcome 4 Estimate the product by rounding to one nonzero digit: 369 x 112. 400 x 100 = 40,000. Check by multiplying a second time. Exact product: 369 x 112 = 41,328. Section 1-5: Dividing Whole Numbers and Decimals Learning Outcome 1 Write 20 divided by 5 in four ways. ) 20 ‚ 5, 5 20, 20 , 20 5 5 Learning Outcome 2 21 R 8 22 470 ) 44 30 22 8 47 ÷ 22 = 2. Place the 2 in the product over the 7 of the minuend. 2 × 22 = 44; place the 44 under the 47 and subtract. 47 – 44 = 3. Bring down the 0 from the minuend to make the 3 a 30. 30 ÷ 22 = 1. Place the 1 in the product over the 0 of the minuend. 1 × 22 = 22; place the 22 under the 30 and subtract. 30 – 22 = 8. 8 is the remainder. Learning Outcome 3 Divide: 14.95 ÷ 2.3 6.5 23. 149.5 ) 138 115 115 000 Move decimal in the divisor one place to the right. Move decimal in dividend the same number of places, that is, 1 place. Place decimal in quotient directly above decimal in dividend. Then divide as with whole numbers. Divide 30.5 ÷ 12 and round to tenths. ) 12 2.54 » 2.5 30.50 24 65 60 50 48 2 Since divisor is a whole number, no decimals are moved. Place decimal in quotient directly over decimal in dividend. Divide one place past tenths place. Since 4 is less than 5, round 2.54 down to 2.5. Divide 12.7 ÷ 6 to tenths and express remainder as a fraction. 2 .1 ) 1 6 6 12.7 12 07 6 1 Since divisor is a whole number, no decimals are moved. Place decimal in quotient directly over decimal in dividend. Divide to tenths place. Write remainder, 1, over the divisor, 6, to express remainder as a fraction. Learning Outcome 4 Estimate 1,875 ÷ 36. Round divisor and dividend each to a number with one nonzero digit, then divide to find first digit of quotient. Add zero(s). 50 40 2, 000 Exact quotient = 52Re. Check: 52 × 36 = 1,872; 1,872 + 3 = 1,875 Divide: 455 ÷ 25 Basic or scientific calculator steps: 455 [÷] 25 [=] Graphing calculator steps: 455 [÷] 25 [EXE] Quotient = 18.2. Learning Outcome 5 Find the average of 78, 85, and 96. 78 + 85 + 96 = 259 259 ÷ 3 = 86.33 or 86 (rounded) Multiply quotient times divisor. Add the remainder. The result should equal original dividend. Add the values. Divide by the number of values. Section 1-6: Exponents, Roots, and Powers of 10 Learning Outcome 1 34 = 3 × 3 × 3 × 3 = 81 81 = 8 60 = 1 3 is a factor 4 times. a1 = a. a0 = 1. Learning Outcome 2 82= 64 64 = 8 8 × 8 = 64. Since 82 is 64, 8 is the principal square root of 64. Use a calculator to find 112, 153, 65, and 4.54. 112 11[x2] ⇒ 121 or 11[xy] 2 = ⇒ 121 153 15[x3] ⇒ 3375 or 15[xy] 3 = ⇒ 3375 65 6[xy] 5 [=]⇒ 7776 4.54 4.5[xy] 4 [=]⇒ 410.0625 Use a calculator to find 529 and 529 [ ] 529 = ⇒ 23 20.25 [ ] 20.25 = ⇒ 4.5 20.25 . [ENTER] or [EXE] may substitute for [=] Learning Outcome 3 Multiply: 18 × 104= 180,000 25 × 100 = 2500 21.55 × 1,000 = 21,550 Divide: 4.65 ÷10 = 0.465 1650 ÷103= 1.650 45,000 ÷104= 4.5 Move decimal 4 places to the right. Move decimal 2 places to the right. Move decimal 3 places to the right. Move decimal 1 place to the left. Move decimal 3 places to the left. Move decimal 4 places to the left. Section 1-7: Order of Operations and Problem Solving Learning Outcome 1 42+ 3(6 – 2) 2 42+ 3(4) 2 16 + 3(4) 2 16 + 12 2 16 + 6 22 Do subtracting in parentheses first. Do exponential operations next. Do multiplication. Do division. Do addition. 42 + 3(6 – 2) 2 Calculator: 4 [x2] [+] 3 [x] [(] 6 [–] 2 [)] [÷] 2 [=]