Decimals and Decimal Applications

For ACT Students
The ACT is a timed exam...60 questions for 60 minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for any wrong answer.

For JAMB and CMAT Students
Calculators are not allowed. So, the questions are solved in a way that does not require a calculator.

Solve all questions.
Use at least two methods (two or more methods) whenever applicable.
Check your work whenever applicable.
Show all work.

(1.) CSEC Using a calculator, or otherwise, determine the EXACT value of:

$ (ii)\:\: (4.14 \div 5.75) + (1.62)^2 \\[3ex] (iii)\:\: 2 * 3.142 * 1.25 \\[3ex] $

$ (ii) \\[3ex] (4.14 \div 5.75) + (1.62)^2 \\[3ex] = 0.72 + 2.6244 \\[3ex] = 3.3444 \\[3ex] (iii) \\[3ex] 2 * 3.142 * 1.25 \\[3ex] = 7.855 $

(2.) CSEC Using a calculator, or otherwise, evaluate EACH of the following, giving your answers to two decimal places.

$ (i)\:\: 73.18 - 5.23 * 9.34 \\[3ex] (ii)\:\: \dfrac{3.1^2}{6.17} + 1.12 \\[5ex] $

$ PEMDAS \\[3ex] (i) \\[3ex] 73.18 - 5.23 * 9.34 \\[3ex] = 73.18 - 48.8482 \\[3ex] = 24.3318 \\[3ex] \approx 24.33\:\:to\:\:two\:\:decimal\:\:places \\[3ex] (ii) \\[3ex] \dfrac{3.1^2}{6.17} + 1.12 \\[5ex] = \dfrac{9.61}{6.17} + 1.12 \\[5ex] = 1.55753647 + 1.12 \\[3ex] = 2.67753647 \\[3ex] \approx 2.68\:\:to\:\:two\:\:decimal\:\:places $

(3.) (a.) The diameter of the planet of Mars is approximately 6.8 × 10³ kilometers.
Express this number in standard notation.

(b.) Paul is an astronomer.
His infrared telescope is able to detect radiation with a wavelength of 0.00000384 meters.
Express this number in scientific notation.

$ (a.) \\[3ex] Diameter\;\;of\;\;Mars \\[3ex] \approx 6.8 * 10^3 \\[3ex] \approx 6.8 * 1000 \\[3ex] \approx 6800\;km ...standard\;\;notation \\[3ex] (b.) \\[3ex] Wavelength \\[3ex] = 0.00000384 \\[3ex] = 3.84 * 10^{-6}\;m ...scientific\;\;notation $

(4.) The United States has seen the federal debt grow considerably and the total debt now is over $17 trillion dollars.
On a single day in October of 2013, the debt grew by a record amount in a single day when the federal government borrowed $328 billion dollars to fund measures from the past five months.
The previous record was set two years ago when $238 billion dollars was added to the national debt.
(Source: Dinan, S.: U.S. debt jumps a record $328 billion - tops $17 trillion for the first time.: The Washington Times. 18 Oct 2013)

Convert the U.S national debt to scientific notation. Choose the correct answer below.

$ A.\;\; 1.7 * 10^{12} \\[3ex] B.\;\; 17 * 10^{12} \\[3ex] C.\;\; 1.7 * 10^{13} \\[3ex] D.\;\; 17 * 10^{14} \\[3ex] $

$ 17\;\;trillion \\[3ex] = 17 * 10^{12} \\[3ex] = 1.7 * 10 * 10^{12} \\[3ex] = 1.7 * 10^{13} $

(5.) Round the following numbers to the nearest integer.
(a.) 6.19
(b.) 341.275
(c.) 12,333.5
(d.) −17.2805

Nearest integer means that we should round to an integer
Look at the first number after the decimal point.
If it is less than 5, discard it, the decimal point, and any remaining numbers.
Just write only the number(s) before the decimal point.
If it is 5 or more, call it 1 and add it to the last number before the decimal point.
Discard the decimal point and any remaining number(s)
Then, write all the number(s) before the decimal point.

To the nearest integer:
6.19 1 < 5 Discard .19 ≈ 6	341.275 2 < 5 Discard .275 ≈ 341	12,333.5 5 = 5 Call it 1; add to 3 from behind 3 + 1 = 4 ≈ 12,334	−17.2805 2 < 5 Discard .2805 ≈ −17

(6.) Which of the following numbers has the most significant digits?
1.08 1.080 0.000080

$ \underline{Significant\;\;Digits} \\[3ex] 1.08 \;\;has\;\; 3\;s.d \\[3ex] 1.080 \;\;has\;\; 4\;s.d \\[3ex] 0.000080 \;\;has\;\; 2\;s.d \\[3ex] $ 1.080 has the most significant digits.
The number 1.080 has the most significant digits because there are four significant digits.

(7.) Express these decimals as simplified fractions.
(a.) 6.333...

There are at least two approaches that we can use to convert repeating decimals to fractions.
Use any approach you prefer.

$ \underline{1st\;\;Approach:\;\; Algebra} \\[3ex] Let\;\;x = fraction\;\;equivalent\;\;of\;\;6.333... \\[3ex] x = 6.33\bar{3}...eqn.(1) \\[3ex] 1\;\;value\;\;before\;\;the\;\;decimal\;\;point \implies \\[3ex] Multiply\;\;both\;\;sides\;\;by\;\;10 \\[3ex] 10 * x = 10 * 6.33\bar{3} \\[3ex] 10x = 63.3\bar{3}...eqn.(2) \\[3ex] Subtract\;\;eqn.(1)\;\;from\;\;eqn.(2) \implies \\[3ex] eqn.(2) - eqn.(1) \implies \\[3ex] 10x - x = 63.3\bar{3} - 6.33\bar{3} \\[3ex] 9x = 57 \\[3ex] x = \dfrac{57}{9} \\[5ex] x = \dfrac{19}{3} \\[5ex] \underline{2nd\;\;Approach:\;\;Arithmetic} \\[3ex] Based\;\;on\;\;the\;\fact\;\;that\;\;0.99\bar{9} = 1 \\[3ex] 6.33\bar{3} = \dfrac{6.33\bar{3}}{1} \\[5ex] = \dfrac{18.99\bar{9}}{3} \\[5ex] = \dfrac{18 + 0.99\bar{9}}{3} \\[5ex] = \dfrac{18 + 1}{3} \\[5ex] = \dfrac{19}{3} $

(9.) Round the following numbers to the nearest tenth.
(a.) 196.35
(b.) 412.143
(c.) 412.963
(d.) 661.9
(e.) −59.714

Nearest tenth is the same as one decimal place
So, we shall be looking at the second number after the decimal place.
If it is less than 5, we discard it and any remaining one(s)
If it is 5 or more, we call it 1 and add it to the first number after the decimal place.

To the nearest tenth:
196.35 5 = 5 Call it 1; add to 3 1 + 3 = 4 ≈ 196.4	412.143 4 < 5 Discard 43 ≈ 412.1	412.963 6 > 5 Call it 1 and add to 9 This gives 10. Call it 1 again and add to 2 ≈ 413.0	661.9 There is no second number after the decimal point Leave as is = 661.9	−59.714 1 < 5 Discard 14 ≈ −59.7

(13.) Round the following numbers to the nearest ten.
(a.) 109.13
(b.) 412.963
(c.) 685.7
(d.) −72.548

Nearest ten means that we should first look at the tens digit of the number.
Then, we look at the unit digit.
If the unit digit (the next number after the tens digit) is less than 5, call it 0 and discard any remaining decimal and number(s)
If the unit digit is 5 or more, call it 1 and add it to the tens digit. Then write 0 as the unit digit and discard any remaining decimal and number(s)

To the nearest ten:
109.13 unit digit = 9 9 > 5 Call it 1; add to 0 1 + 0 = 1 Then, write 0 as the unit digit and discard .13 ≈ 110	412.963 unit digit = 2 2 < 5 Call it 0 and discard .963 ≈ 410	685.7 unit digit = 5 5 = 5 Call it 1; add to 8 1 + 8 = 9 Then, write 0 as the unit digit and discard .7 ≈ 690	−72.548 unit digit = 2 2 < 5 Call it 0 and discard .548 ≈ −70

(17.) Perform these arithmetic operations and approximate the answer to the stated number of significant digits (s.d).
(a.) 42 * 32.7; 3 significant digits
(b.) 259.82 ÷ 0.092; 2 significant digits
(c.) $(1.68 * 10^3) * (6.6 * 10^{-2})$; 3 significant digits

To convert to significant digits (significant figures), it is important to know the Rules for Counting Significant Digits.

To the specified number of significant digits:
(a.) 42(32.7) = 1373.4 1373.4 has 5 significant digits (Rule #1) To approximate it to 3 significant digits: Look at the 4th number The 4th number is 3 3 < 5 Call it 0 Discard .4 ≈ 1370 to 3 s.d	(b.) 259.82 ÷ 0.092 = 2824.130435 2824.130435 has 10 s.d (Rules #1 and #2) To approximate it to 3 significant digits: Look at the 3rd number The 3rd number = 2 2 < 5 Call it 0. Call the 4th number 0. Discard .130435 ≈ 2800 to 2 s.d
$ (c.) \\[3ex] (1.68 * 10^3) * (6.6 * 10^{-2}) \\[3ex] = 1.68 * 6.6 * 10^3 * 10^{-2} \\[3ex] = 11.088 * 10^{3 + -2} \\[3ex] = 11.088 * 10^{3 - 2} \\[3ex] = 11.088 * 10^1 \\[3ex] = 1.1088 * 10^1 * 10^1 \\[3ex] = 1.1088 * 10^{1 + 1} \\[3ex] = 1.1088 * 10^2 \\[3ex] $ Let us focus on 1.1088 1.1088 has 5 significant digits (Rule #2) To approximate it to 3 significant digits: Look at the 4th number 8 > 5 Call it 1 and add to 0 1 + 0 = 1 Discard the rest This becomes: 1.11 ≈ 1.11 * 10^2

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