KNOW YOUR NUMBERS!

Factors, Multiples, Fractions, Percents, Estimates, Change

From a Dollar, Tips

FRACTIONS, DECIMALS, PERCENTS, & ANGLES

Convert from one to any of the other three… with or

without a calculator!

Purpose

• Know your numbers! In this case, understand how to convert fractions to decimals, percents, or pieces of a circle.

• Get a “feel” for each. Estimate a fraction based on the decimal. Create a circle graph given a percent or a fraction.

• Circle has 360 degrees. Multiply fraction, decimal, or percent times 360.

Fraction – Decimal – Percent - Degrees

• Fraction Decimal Percent Degrees• ½ 0.5 50% 180°• 0.3333… 33 % 120°• 0.6666… 66 % 240°• ¼ 0.25 25 % 90°• ¾ 0.75 75 % 270° • YOU SHOULD MEMORIZE THESE!!!• (in other words, no calculator needed!)

Fraction – Decimal – Percent - Degrees

• 1/5 0.2 20% 72°• 2/5 ____ ____ % ___°• 3/5 ____ ____ % ___°• 4/5 ____ ____ % ___°• 1/6 ____ ____ % ___°• 5/6 ____ ____ % ___°• Why didn’t I include 2/6, 3/6, and 4/6?• Simplified: 1/3, ½, and 2/3. You have ‘em!

Fraction – Decimal – Percent - Degrees

• 1/5 0.2 20% 72°• 2/5 0.4 40 % 144°• 3/5 0.6 60 % 216°• 4/5 0.8 80 % 288°• 1/6 0.1666… 16 % 60°• 5/6 0.8333… 83 % 300°

Fraction – Decimal – Percent - Degrees

7 is not a factor of 360, so round the percents and degrees to 1 decimal place. Write decimal to six places, however.

• 1/7 ______ _____ % ____°• 2/7 ______ _____ % ____°• 3/7 ______ _____ % ____°• 4/7 ______ _____ % ____°• 5/7 ______ _____ % ____°• 6/7 ______ _____ % ____°• What patterns do you notice?

Fraction – Decimal – Percent - Degrees

• 1/7 0.142857… 14 % 51.4°• 2/7 0.285714… 28 4/7 % 102.9°• 3/7 0.428571… 42 6/7 % 154.3°• 4/7 0.571428… 57 % 205.7°• 5/7 0.714285… 71 3/7 % 257.1°• 6/7 0.857142… 85 5/7 % 308.6°• Each decimal is the same six-digit repeater,

beginning at different places.

Fraction – Decimal – Percent - Degrees

• 1/8 _____ _____ % ____°• 3/8 _____ _____ % ____°• 5/8 _____ _____ % ____°• 7/8 _____ _____ % ____°• Again, what patterns do you notice? • Do the ninths (1/9, 2/9, etc.) and find the easy

pattern for them.

Fraction – Decimal – Percent - Degrees

• 1/8 0.125 12½ % 45°• 3/8 0.375 37½ % 135°• 5/8 0.625 62½ % 225°• 7/8 0.875 87½ % 315°• 1/9 0.111… 11 1/9 % 40°• 2/90.222… 22 2/9 % 80°• Tenths are easy. 1/10 = 0.1 = 10%, and

so on (2/10 = 0.2 = 20%, etc).

F – D – P – D

• 1/11 0.090909… 9 1/11 % 32.7°

• 2/11 0.181818… 18 2/11 % 65.5°

• 3/11 0.272727… 27 3/11 % 98.2°

• You should be able to see the pattern for the remaining elevenths.

• Now you do the twelfths. • You should be able to convert any fraction

to a decimal or percent.

Converting Back & Forth

• To convert any fraction or decimal to a percent, multiply by 100.

• To convert any percent to a fraction or decimal, divide by 100.

• abcDefghijklmnoPqrstuvwxyz• Decimal to Percent . 2 places• Percent to Decimal . 2 places

Convert Each One

• Fraction Decimal Percent• 13 / 16 ______ _____ %• _____ 0.3125 _____ %• _____ _____ 41 %• Try these to show you’re a genius:• _____ 0.1444444… 144/9 %

• _____ 0.633333.. 63 %

Convert Each One

• Fraction Decimal Percent• 13 / 16 0.8125 81¼ %• 5 / 16 0.3125 31¼ %• 5 / 12 0.416666… 41 %• Are You a genius?• 13 / 90 0.1444444… 144/9 %

• 19 / 30 0.633333.. 63 %

Estimate the Fraction• Suppose you have a percent, and need to

“think” of it as a relatively simple fraction, involving halves, thirds, fourths, fifths, etc.

• Round the percent to one that matches a fraction that you know.

• Example: 43.7% is a tad more than 40%, which is 2/5. So, if 43.7% voted for Candidate X, then about 2 out of every 5 voters voted for him.

Estimate the Fraction

• 35% is a bit more than ….• One third• 72% is just a hair less than…• Three fourths• 83% is almost exactly…• Five sixths• 60% is exactly…• Three fifths

Estimate the Fraction

• 38.7%• About 2/5th • 13%• About 1/8th • 14.5%• About 1/7th • 17%• About 1/6th

Tip? • Granted, most of you don’t even think

about tipping the server (waiter/waitress), but most adults do!

• “Standard” tip is about 15 to 18%. • Think about these fractions & percents:• 1/7 = 14.3% 1/6 = 16.7% 1/5 = 20% • What would you tip for average, good, or

super service on a $21.97 bill?

Tip on $21.97?

• If you tip $1 for every $7 you spend, that’s a bit more than 14%. For every $6 you spend, that’s almost 17%. For every $5 you spend, that’s 20%.

• A $3 tip (3 × 7 = 21) is less than 15%, but a $4 tip is about 18%.

• Round the bill to the nearest multiple of 7, 6, or 5, based on your tipping percentage.

Round the Bill• Bill 7x (14.3%) 6x (16.7%) 5x (20%) • $34.83 $35 $36 $35• $17.12 $14 $18 $15• $22.75 ____ ____ ____• $47.29 ____ ____ ____• $27.33 ____ ____ ____• $8.93 ____ ____ ____• What would you tip in each case?

Round the Bill • Bill 7x (14.3%) 6x (16.7%) 5x (20%) • $34.83 $35 $36 $35• $17.12 $14 $18 $15• $22.75 $21 $24 $20/25• $47.29 $49 $48 $45/50• $27.33 $28 $24/30 $25/30• $8.93 $7 $6/12 $5/10• What would you tip in each case?

Round the Bill • Bill (14.3%) (16.7%) (20%) • $34.83 $5 $6 $75• $17.12 $2 $3 $3 or 4• $22.75 $3 $4 $4 or 5• $47.29 $7 $8 $9 or

10• $27.33 $4 $4 or 5 $5 or 6• $8.93 $1 $1 or 2 $1 or 2• Tip is rounded to nearest dollar.

Tip: Cash or Plastic?

• If you’re leaving cash on the table, you probably leave whole dollars, or maybe dollars and a couple of quarters.

• But, if you’re using plastic, what’s typical?• Suppose bill is $23.71. If you’re tipping a

dollar for every $6 you spend, that’s about $4. (24 is a multiple of 6) So, tip is $4.29, which brings total bill to $28.00

Change From A Dollar

• If you’re paying in cash, don’t you want correct change? Of course! What’s the change back from a dollar :

• $.53 $.82 $.67 $.39• $.47 $.18 $.33 $.61• Can you do this quickly, in your head?• $.34 $.29 $.58 $.74• $.66 $.71 $.42 $.26

PERCENTS

Solve Percent Problems

Three Typical Problems

• What number is 20% of 30? x• 30 is 20% of what number? y• 20 out of 30 is what percent? z• All three have 20 & 30, but different

answers. • Answers: x = 6 y = 150 z = 66%• Note that only one answer has a %

Percent Proportion

• Use this proportion to solve percent problems.

• IS Percent• = • OF 100

Solving Percent Problems• Use the “IS over OF equals Percent Over

One Hundred” proportion.• When the problem calls for a percent of

increase (growth) or decrease (decline), use “Difference” as the “IS” and “OLD” as the “OF.”

• Difference = New – Old.

Solve

• What is 33 1/3 % of 123? • Hint: 33 1/3% is one-third! Just divide by

3, or, set up proportion: is / of = 1 / 3• ? 1 • ---- = ----- • 123 3 • 41

Solve

• 18 correct on a 22 question quiz is what percent correct?

• 18 ?• ------ = ------• 22 100 • 18 × 100 ÷ 22= ?• 819/11 % (81.8, rounded to nearest tenth)

Solve• 13,500 voters represent 25% of the

eligible voting population. How many people are eligible to vote?

• 13,500 = “IS” 25% out of 100 is ¼ , so you could just multiply by 4.

• 13,500 / x = 25 / 100 13,500 / x = 1 / 4• Total population (the “OF”) = 54,000.

As Seen on TV…• “Four out of five doctors” recommend

Tylenol to their patients. • What percent is that? • IS = 4 OF = 5 4 / 5 = % / 100• 80% • By the way, the makers of Tylenol give

loads of free samples to doctors, so they give ‘em out! Thus, the ad…

Percent of Change

• (New – Old) Percent of Change• ----------------- = ------------------------• Old 100 • It’s the same “IS over OF” ratio, but the

“IS” is the difference between the old value and the new one.

• Old…Original...Of (How to remember?)

Percent of Change

• In 2013-2014, we had about 1320 students. Next year, it was about 1400. What percent of change is that?

• Step 1: Find the difference. 1400 – 1320 is an increase of 80 students.

• Step 2: Use “Diff / Old = % / 100” proportion, cross-multiplying & dividing.

• 100 × 80 ÷ 1320 ≈ 6.1% increase

“B O G O”

• Many stores have “BOGO” sales. “Buy One, Get One” or sometimes “Buy Two, Get One,” and so on. What percent do you save?

• Example: “Buy one, get one free” is a 50% savings, since 1 free out of 2 is ½.

• Ex: “Buy two, get one free” is a 33 1/3% savings. (1 out of 3 is free)

BOGO

• Suppose you see: “Buy one, get second half off.” What percent is savings?

• Think: If you had to pay full price for two, that’s the equivalent of 200%. You pay 100% for the first one, but 50% for the second. So you pay 150% vs. 200%.

• 200 – 150 = 50. You save 50 out of 200• 50 / 200 = 0.25 You save 25%

Which is Cheaper?

• Macy’s is having a sale, where everything in the store is 15% off. Their regular price of a pair of shoes is $44.95.

• Dillards is having a sale on those same shoes, which are discounted 20%. Their regular price is $48.95.

• J C Penney sells the same shoes at $38.95.

• Which store has the cheapest shoes?

Where are the Cheapest Shoes?• Macy’s: 15% × $44.95 = $6.74 Subtract

to get price of $38.21. • Dillards: 20% × 48.95 = $9.79 Subtract to

get price of $39.16 Dillards costs more.• All prices within a dollar and change of

each other, Macy’s slightly cheaper. • Alternate method: 100% – 15% = 85%

Multiply $44.95 × 85% = $38.21