Question

e. $(103.05 + 16.9)/(11.13 - 2.6) = \underline{\quad\quad\quad\quad\quad\quad\quad}$

1. dimensional analysis (unit conversions)

core idea: use conversion factors to cancel units until only the desired unit remains

• conversion factor: a ratio equal to 1 (exact relationship), written in two ways.
• steps (one factor): identify given/needed units $\to$ write the conversion factor $\to$ set up so units cancel $\to$ calculate $\to$ round to correct sig figs.
• steps (multiple factors): use an intermediate unit and multiply conversion factors so unwanted units cancel.

practice: show setup with units and cancel units.
a) 150 minutes = $\underline{\quad\quad\quad\quad\quad\quad\quad}$ hours
b) 0.750 l = $\underline{\quad\quad\quad\quad\quad\quad\quad}$ ml
c) 753 cg = $\underline{\quad\quad\quad\quad\quad\quad\quad}$ mg
d) 772 g = $\underline{\quad\quad\quad\quad\quad\quad\quad}$ kg
e) 420 cm = $\underline{\quad\quad\quad\quad\quad\quad\quad}$ m
f) 0.0084 ml = $\underline{\quad\quad\quad\quad\quad\quad\quad}$ $\mu$l
g) 955 cm = $\underline{\quad\quad\quad\quad\quad\quad\quad}$ km
h) which is longer: 1.1 km or 520,000 cm? $\underline{\quad\quad\quad\quad\quad\quad\quad}$
i) volume of a cube (2 cm on each side) in ml: $\underline{\quad\quad\quad\quad\quad\quad\quad}$
1 - minute wrap - up

• sig figs communicate precision: all certain digits + one estimated digit.
• rounding rules: 0–4 down, 5–9 up (based on the next digit).
• zeros may be significant or placeholders—use the rules, especially for trailing zeros.
• exact numbers have unlimited sig figs.
• calculations: $\times / \div \to$ fewest sig figs; $+ / - \to$ fewest decimal places.
• dimensional analysis: conversion factors cancel units; units guide the setup

exit ticket (1 minute)

1. which digit in a measurement is always uncertain? $\underline{\quad\quad\quad\quad\quad\quad\quad}$
2. true/false: leading zeros (e.g., 0.0045) are significant. t / f
3. true/false: 1 ml = 1 cm³. t / f
4. write one sentence: what do significant figures tell you?

Explanation:

Step1: Calculate top/bottom parentheses

Top: $103.05 + 16.9 = 119.95$
Bottom: $11.13 - 2.6 = 8.53$

Step2: Divide top by bottom

$\frac{119.95}{8.53} \approx 14.06$
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A) Step1: Use 60 min = 1 hr

$150 \text{ min} \times \frac{1 \text{ hr}}{60 \text{ min}}$

A) Step2: Cancel & calculate

$\frac{150}{60} = 2.5$
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B) Step1: Use 1 L = 1000 mL

$0.750 \text{ L} \times \frac{1000 \text{ mL}}{1 \text{ L}}$

B) Step2: Cancel & calculate

$0.750 \times 1000 = 750$
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C) Step1: Use 1 cg = 10 mg

$753 \text{ cg} \times \frac{10 \text{ mg}}{1 \text{ cg}}$

C) Step2: Cancel & calculate

$753 \times 10 = 7530$
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D) Step1: Use 1 kg = 1000 g

$772 \text{ g} \times \frac{1 \text{ kg}}{1000 \text{ g}}$

D) Step2: Cancel & calculate

$\frac{772}{1000} = 0.772$
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E) Step1: Use 1 m = 100 cm

$420 \text{ cm} \times \frac{1 \text{ m}}{100 \text{ cm}}$

E) Step2: Cancel & calculate

$\frac{420}{100} = 4.20$
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F) Step1: Use 1 mL = 1000 μL

$0.0084 \text{ mL} \times \frac{1000 \text{ μL}}{1 \text{ mL}}$

F) Step2: Cancel & calculate

$0.0084 \times 1000 = 8.4$
---

G) Step1: Use 1 km = 100000 cm

$955 \text{ cm} \times \frac{1 \text{ km}}{100000 \text{ cm}}$

G) Step2: Cancel & calculate

$\frac{955}{100000} = 0.00955$
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H) Step1: Convert 1.1 km to cm

$1.1 \text{ km} \times \frac{100000 \text{ cm}}{1 \text{ km}} = 110000 \text{ cm}$

H) Step2: Compare values

$110000 \text{ cm} < 520000 \text{ cm}$
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I) Step1: Calculate cube volume

$V = (2 \text{ cm})^3 = 8 \text{ cm}^3$

I) Step2: Convert cm³ to mL

$8 \text{ cm}^3 \times \frac{1 \text{ mL}}{1 \text{ cm}^3} = 8$
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Exit 1) Step1: Recall sig fig rules

The last digit in a measurement is estimated, so it is uncertain.

Exit 2) Step1: Recall leading zero rule

Leading zeros are placeholders, not significant.

Exit 3) Step1: Recall volume equivalence

$1 \text{ cm}^3$ is defined as $1 \text{ mL}$.

Exit 4) Step1: Define sig fig purpose

Significant figures communicate the precision of a measured value.

Answer:

E. $14.1$ (rounded to 3 sig figs)
A) $2.5$ hours
B) $750$ mL
C) $7530$ mg
D) $0.772$ kg
E) $4.20$ m
F) $8.4$ μL
G) $0.00955$ km
H) $520,000$ cm
I) $8$ mL

Exit Ticket:

1. The final/estimated digit
2. F (False)
3. T (True)
4. Significant figures tell you the precision of a measured or calculated value.