Question

1. round the following values as indicated in the table below: (4 marks)

	round to 3 sig figs	round to 2 decimal places
0.007765	(handwritten entry)	(blank)
1.0006	(blank)	(blank)
0.065	(blank)	(blank)

Explanation:

For 86,511.83450:

Step1: Round to 3 Sig Figs

Significant figures start from the first non - zero digit. For 86511.83450, the first three significant figures are 8, 6, 5. The next digit is 1, which is less than 5, so we round down. So 86511.83450 rounded to 3 significant figures is $86500$ (in scientific notation, it can be written as $8.65\times10^{4}$, but in the context of the number, we can present it as 86500).

Step2: Round to 2 Decimal Places

We look at the third decimal place. The number is 86511.83450. The third decimal place is 4, which is less than 5. So we keep the second decimal place as it is. So 86511.83450 rounded to 2 decimal places is $86511.83$.

For 0.007765:

Step1: Round to 3 Sig Figs

The first non - zero digit is 7 (the first 7 after the leading zeros). So the significant figures are 7, 7, 6 (the third significant figure). The next digit is 5, so we round up the third significant figure. So 0.007765 rounded to 3 significant figures is $0.00777$.

Step2: Round to 2 Decimal Places

The number 0.007765 has two decimal places as 0.00 (since the first non - zero digit is in the thousandths place). But if we consider the decimal places, when rounding to 2 decimal places, we look at the third decimal place. The number is 0.007765. The third decimal place is 7. But since the first two decimal places are 0 and 0, and the third decimal place is 7 which is greater than 5, we round up the second decimal place? Wait, no. Wait, 0.007765: the first decimal place is 0 (tenths), second is 0 (hundredths), third is 7 (thousandths). When rounding to 2 decimal places, we look at the digit in the third decimal place (7). Since 7>5, we round up the hundredths place. But the hundredths place is 0, so it becomes 0.01? Wait, no, let's re - calculate. 0.007765: the number is 0.00 (two decimal places) plus 0.007765. Wait, the correct way: the number is 0.007765. To round to 2 decimal places, we look at the third decimal digit (the digit in the thousandths place), which is 7. Since 7 ≥ 5, we add 1 to the second decimal digit. The second decimal digit is 0, so 0 + 1=1. So 0.007765 rounded to 2 decimal places is $0.01$.

For 1.0006:

Step1: Round to 3 Sig Figs

The significant figures start from 1. The first three significant figures are 1, 0, 0. The next digit is 0 (wait, no: 1.0006. The digits are 1 (first sig fig), 0 (second), 0 (third), 0 (fourth), 6 (fifth). Wait, no, significant figures: all non - zero digits and zeros between non - zero digits are significant. So 1.0006 has significant figures 1, 0, 0, 0, 6. To round to 3 significant figures, we look at the fourth significant figure, which is 0. Wait, no, the first three significant figures are 1, 0, 0. The next digit is 0 (the fourth digit after 1). Wait, no, 1.0006: the number is 1.0006. The first significant figure is 1, second is 0, third is 0, fourth is 0, fifth is 6. Wait, no, I made a mistake. The correct way: for 1.0006, the significant figures are 1, 0, 0, 0, 6. To round to 3 significant figures, we look at the digit after the third significant figure. The third significant figure is 0 (the third digit: 1.00 06). The digit after the third significant figure is 0 (the fourth digit), which is less than 5? Wait, no, 1.0006: let's write it as 1.0006. The first three significant figures are 1, 0, 0. The next digit is 0 (the fourth digit), so we keep the third significant figure as it is? Wait, no, that's not right. Wait, 1.0006: the number is 1.0006. To round to 3 significant figures, we look at the 4th digit (the digit in the ten - thousandths place? No, in terms of significant figures, the positions are: 1 (1st), 0 (2nd), 0 (3rd), 0 (4th), 6 (5th). Wait, no, the decimal is after 1, so 1.0006 is 1 + 0.0006. The significant figures are 1, 0, 0, 0, 6. So when rounding to 3 significant figures, we look at the digit after the third significant figure (the 4th significant figure, which is 0). Since 0 < 5, we round down. So 1.0006 rounded to 3 significant figures is $1.00$.

Step2: Round to 2 Decimal Places

We look at the third decimal place. 1.0006: the first decimal place is 0 (tenths), second is 0 (hundredths), third is 0 (thousandths), fourth is 6 (ten - thousandths). Wait, no, 1.0006: the decimal places are: 0 (tenths), 0 (hundredths), 0 (thousandths), 6 (ten - thousandths). To round to 2 decimal places, we look at the third decimal place (0). Wait, no, 2 decimal places: the number is 1.00 (two decimal places) plus 0.0006. Wait, the third decimal place is 0, which is less than 5, so we keep the second decimal place as it is? Wait, no, 1.0006: the number is 1.0006. The second decimal place is 0, the third decimal place is 0. Wait, no, I think I messed up. 1.0006: the digits after the decimal are 0 (1st decimal place - tenths), 0 (2nd - hundredths), 0 (3rd - thousandths), 6 (4th - ten - thousandths). So to round to 2 decimal places, we look at the 3rd decimal place (0). Since 0 < 5, we round down. So 1.0006 rounded to 2 decimal places is $1.00$. Wait, but the fourth decimal place is 6, but we are rounding to 2 decimal places, so we only care about the digit in the third decimal place. Wait, no, the rule is: when rounding to n decimal places, we look at the (n + 1)th decimal place. So for 2 decimal places, we look at the 3rd decimal place. In 1.0006, the 3rd decimal place is 0, so we keep the 2nd decimal place as 0. So 1.0006 rounded to 2 decimal places is $1.00$.

For 0.065:

Step1: Round to 3 Sig Figs

The first non - zero digit is 6 (the first significant figure), second is 5 (second), and we need a third significant figure. Since there are only two significant figures in 0.065 (6 and 5), we can write it as 0.0650 (adding a zero at the end, which is a significant figure when we are rounding). Wait, 0.065: the significant figures are 6 and 5. To round to 3 significant figures, we add a zero at the end, so 0.0650.

Step2: Round to 2 Decimal Places

0.065: the first decimal place is 0 (tenths), second is 6 (hundredths), third is 5 (thousandths). To round to 2 decimal places, we look at the third decimal place (5). Since 5 ≥ 5, we round up the second decimal place. The second decimal place is 6, so 6+1 = 7? Wait, no, 0.065: the number is 0.06 (two decimal places) plus 0.005. When rounding to 2 decimal places, we look at the third decimal place (5). So we add 1 to the second decimal place. 6+1 = 7? Wait, no, 0.065: 0.06 (two decimal places) and the third decimal is 5. So 0.06 + 0.01=0.07? Wait, no, 0.065 is equal to 0.065000... When rounding to 2 decimal places, the second decimal place is 6, the third is 5. So we round up the second decimal place: 6 becomes 7? Wait, no, 0.065: let's calculate. 0.065 is 65 thousandths. To two decimal places, which is hundredths. 65 thousandths is 0.065. The hundredths place is 6, the thousandths place is 5. So we round up the hundredths place: 6 + 1=7, so 0.07. Wait, but 0.065 is exactly halfway between 0.06 and 0.07? In some rounding rules, we round to the even number, but usually, for the purpose of this problem, we can round 0.065 to 0.07 when rounding to 2 decimal places.

Answer:

Value	Round to 3 Sig Figs	Round to 2 Decimal Places
0.007765	0.00777	0.01
1.0006	1.00	1.00
0.065	0.0650	0.07