Questions and Answers #10 Decimals

Questions and Answers Sheet 10 Decimals Question #1 The number 1.5 is special because it is equal to one quarter of the sum of its digits, as 1+5=6 and 64=1.5. Find all the numbers that are equal to one quarter of the sum of their own digits. Answer: This is not much better than a brute force method: The sum of the digits S is a non-negative integer, so a quarter of it 𝑆 4 s non-negative, of the form x or x.25 or x.5 or x.75 for some non-negative integer x. We must have x<10 since if 10 ≤ 𝑥 < 100 then 40 ≤ 4𝑥 ≤ 𝑆 < 412 so the sum of the digits of x must be at least 40-12=28 but two-digit integers have sums of digits no more than 18. Similarly with larger x. So a satisfactory x has sum of digits 4x=S=x with solution x=0 and a satisfactory x.25 has sum of digits 1 4 (𝑥 + ) = 𝑆 = 𝑥 + 7 4 with solution x=2 and a satisfactory x.5 has sum of digits 1 4 (𝑥 + ) = 𝑆 = 𝑥 + 5 2 with solution x=1 and a satisfactory x.75 has sum of digits 3 4 (𝑥 + ) = 𝑆 = 𝑥 + 12 4 with solution x=3 making the numbers which work 0, 2.25, 1.5, 3.75 Question #2 The number of decimals numbers between different size number ranges equal? For instance, is it correct to say that the number of decimals Numbers between 220 and 289.999 ⋯ and the number of decimal numbers between 297 and 297.999 ⋯ are equal? Even though the range of the former is 70 times bigger? Answer: Consider the intervals [a, b] and [c, d] The function 𝑓: [𝑎, 𝑏] → [𝑐, 𝑑] given by 𝑓(𝑥) = ( 𝑐−𝑑 𝑎−𝑑 )𝑥 + ( 𝑎𝑑−𝑏𝑐 𝑎−𝑏 ) is a bijection. This means that any two closed intervals are equinumerous. Interestingly, this means all line segments contain the same number of points, irrespective of their lengths. Question #3 There is no three consecutive integers with same Parity of sum of its digits Let D(a) denotes sum of digits of a in decimal. Examples D(49)=4+9=13 Let P(a) denotes parity of a. Example P(2)=0 as even and P(3)=1 as odd. Questions: show that there is no a such that 𝑃(𝐷(𝑎)) = 𝑃(𝐷(𝑎 + 1)) = 𝑃(𝐷(𝑎 + 2)). Answer: We will divide the numbers into two cases- Those ending with 9 and those which are not. Let, us select a number of the form (ending with 9) k9 where 𝑘 ∈ 𝑁. Clearly, the next number is of the form (k+1)0 Now, k be either odd or even. Note, D(k9)=k+9 is of the opposite parity of k (Why?). And for the second number, 𝐷((𝑘 + 1)0) = 𝑘 + 1 + 0 = 𝑘 + 1 which is of opposite parity of k. So, if a number ends with 9, the number and its successive number are of same parity (0, 0) or (1, 1) Now, select a number of the form (not ending with 9) kn where, 𝑘 ∈ 𝑁 and 𝑛 ∈ {1,2, ⋯ ,8}. The next number is of the form k(n+1) Clearly, 𝐷(𝑘𝑛) = 𝑘 + 𝑛 and 𝐷(𝑘(𝑛 + 1)) = 𝑘 + 𝑛 + 1 which are of opposite parity. Now, see if the result you want follows. Question #4 Using long division, write the rational number − Answer: Let us divide 9 by 24 using long division. 9 24 as a terminating decimal? 24 | ( 0.375 9.000 72  180 ) 168 120 120 − and hence − 9 24 = −0.375 Question #5 What is 0.9978+0.325? Answer: When adding decimals it is very important to line up the decimal point and place numbers of the same value. 0.9978 ( + 0.325 = 1.3228 ) If there is no digit shown - treat it as 0. When you add, you now add the digits with the same place value together. so 9 10 9 100 must be above 7 1000 8 10000 3 must be above must be above 10 2 100 and and 5 1000 is alone. Question #6 What is 1.2 × 0.6 ? Answer: The number 1.2 has only one digit which is 2. The same is for 0.6 - the only one digit is 6. Let me multiply each of them by ten: 1.2 × 10 = 12 0.6 × 10 = 6 When we have converted them to the non-digital numbers we can multiply them easily: 12 × 6 = 72 Now, we should come to the start: we had 1.2 and 0.6 which we each multiplied by 10. Since both of them were multiplied by ten, now the result should be divided by 100 in order to obtain a correct number of digits. 72 100 = 0.72 So 1.2 × 0.6 = 0.72 Question #7 How to solve 𝑎/3.2 = .8 ? Answer: So we have: 𝑎/(3.2) = 0.8 𝑎 = 0.8 × (3.2) 𝑎 = 2.56 Question #8 How to write 8 11/100 as a decimal? Answer: To begin with, change this to an incorrect fraction: 11 11  100  11 8+   8  + 100 100  100  100 800 11 800 + 11 811 +   100 100 100 100 8 Since there are two decimal places in the denominator, the decimal representation of this number is: 811  8.11 100 Question #9 How to convert 30/20 to a decimal? Answer: 30/20 is equivalent to 3/2 because the zeros at the end would be useless. 3/2 expressed as a mixed number is 1 1/2 , which as a decimal would be 1.5. Question #10 How to write 2 51/100 as a decimal number? Answer: 251/100 is a mixed fraction. the fraction 51/100 is not a decimal, Divide the numerator by the denominator in order to convert fractions to decimals. numerator = 51 denominator = 100 51/100=0.51 0.51+2=2.51 That means 251/100 = 2.51 When converting fractions into decimal, the equation is n/d n = numerator = 51 d = denominator = 100 51/100=0.51 Converting a mixed fraction into a decimal number is done by using the equation w+n/d w = whole number = 2 n = numerator = 51 d = denominator = 100 2+51/100=2.51 Question #11 37 1/2% is equal to what fraction? Answer: Solution: 37 1/2 % means (75/2)/100 = 75/200=3/8 Question #12 How to convert -7/4 into decimal form? Answer: So solution: −7/4 = −7 ∗ 1/4 = −7 ∗ 0.25 = −1.75