Math SAT Level I - Chapter 3 Fractions, Decimals, and Percents

To increase a number by r%, multiply it by __________.

(1 + r%)

To decrease a number by r%, multiply it by __________.

(1 - r%)

In a jar, 1/2 of the marbles are red, 1/4 are white, and 1/5 are blue. What fraction of the marbles are neither red, white, nor blue?

(1/2)+(1/4)+(1/5) => (10/20)+(5/20)+(4/20) = 19/20 20/20 - 19/20 = 1/20 of the marbles are neither red, white, nor blue.

There are 800 students at Central High School, each of whom takes exactly one of the four science courses. If 1/5 take earth science, 2/5 take biology, and 3/10 take chemistry, how many students take physics?

(1/5)+(2/5)+(3/10) => 2/10 + 4/10 + 3/10 = 9/10 10/10 - 9/10 = 1/10 x 800 = 80 students take physics

Evaluate (1/6) + (1/4)

(1/6) + (1/4) => (2/12) + (9/12) = 11/12

Evaluate (2/3a) + (3/2a)

(2/3a) + (3/2a) => (4/6a) + (9/6a) => 13/6a

If 2/7 of the 840 students at Monroe High School are freshmen and if 30% of the freshmen play musical instruments, how many freshmen play musical instruments?

(2/7) x 840 = 240 freshmen. Of these 30% play an instrument: 240 x (30/100) = 72

Evaluate (2x+3)/(x^2) + (x+1)/(x^2)

(2x+3)/(x^2) + (x+1)/(x^2) => (3x +4)/(x^2)

Evaluate (2x+3)/(x^2) - (x+1)/(x^2)

(2x+3)/(x^2) - (x+1)/(x^2) => (x+2)/(x^2)

Evaluate (3/4) x π

(3/4) x π => (3/4) x (π/1) => 3π/4

Evaluate (3/5) x 7

(3/5) x 7 => (3/5) x (7/1) => 21/5

The percent decrease of a quantity is?

(the actual decrease)/(the original amount) x 100%

The percent increase of a quantity is?

(the actual increase)/(the original amount) x 100%

If 0 < x < 1, which of the following could be less than x? I. x² II. 10% of x III. 1/x A. I only B. II only C. III only D. I and II only E. I and III only

0 < x < 1 Use a number between 0 and 1. (e.g. 0.5) 0.5² = 0.25 [ I is true ] 0.10(0.5) = 0.05 [ II is true ] 1/0.5 = 2 [ III is false ] D. Only I and II are true

Determine the sign ( < , >, =): 2/3 _____ 5/8

2/3 > 5/8 because (2 x 8) > (3 x 5) 0.666 > 0.625

Evaluate 20 ÷ 2/3

20 ÷ 2/3 => 20/1 x 3/2 => 30

Express as a decimal and a fraction 20%

20% = 0.20 = 20/100 => 1/5

Express as a decimal and a fraction 3%

3% = 0.03 = 3/100

What fraction of 72 is 3/5 of 80? A. 1/3 B. 2/5 C. 5/9 D. 3/5 E. 2/3

3/5 of 80 = 48 What fraction of 72 is 48? 72 x 1/3 = 24 WRONG 72 x 2/5 = 28.8 WRONG 72 x 5/9 = 40 WRONG 72 x 3/5 = 43.5 WRONG 72 x 2/3 = 48 CORRECT

Evaluate 3/5 x 4/7

3/5 x 4/7 => 3 x 4/ 5 x 7 => 12/35

Evaluate 3/5 ÷ 2/3

3/5 ÷ 2/3 => 3/5 x 3/2 = 9/10

Evaluate 4/9 - 1/3

3/9 => 1/3

Express as a decimal and a fraction 350%

350% = 3.50 = 350/100 => 7/2

Express as a decimal and a fraction 37.5%

37.5% = 0.375 = 37.5/100 => 375/1000 => 3/8

Evaluate 4/9 + 1/9

5/9

If 5x = 3 and 3y = 5, what is the value of x/y?

5x = 3 => x = 3/5 3y = 5 => y = 5/3 x/y = (3/5) / (5/3) => 3/5 x 3/5 => 9/25 A. 9/25

If an initial quantity A increases r% per year, the the amount at the end of t years is given by _______________.

A(t) = A(1 + r%)^t

From 1990 to 2000, the number of students attending McKinley High School increased by 3.5% each year. If the school had 800 students in 1990 how many students did it have in 2000? A. 1,080 B. 1,090 C. 1,128 D. 1,168 E. 1,215

A(t) = A(1 + r%)^t The number of students in the school in 2000 was (1.035)¹⁰(800) = 1,128 C. 1,128

If the population of Centerville (which was 10,000 on January 1, 1990) grew at a rate of 5% per year for 10 years, what was its population on January 1, 2000?

A(t) = A(1 + r%)^t The population after 10 years was: A(10) = 10,000( 1 + 0.05)¹⁰ => 10,000(1.6289) =? 16,289

What percent of 50 is w? A. 2w B. 2/w C. 50/w D. w/2 E. w/50

A. 2w w = (x/100)(50) = x/2 => x = 2w

Which of the following list the numbers 3/8, 4/11, and 5/13 in increasing order? A. 3/8, 4/11, 5/13 B. 4/11, 3/8, 5/13 C. 5/13, 4/11, 3/8 D. 4/11, 5/13, 3/8 E. 3/8, 5/13, 4/11

B. 4/11, 3/8, 5/13 4/11 = 0.3636..., 3/8 = 0.375, 5/13 = 0.3846...

54 is 45% of what number?

Call the unknown x. Translate the sentence, replacing "is" by "=" and the unknown by x. is/of = % / 100 54/x = 45/100 => 5400 = 45x => x =120

77 is what percent of 140?

Call the unknown x. Translate the sentence, replacing "is" by "=" and the unknown by x. is/of = % / 100 77/140 = x/100 => 140x = 7700 => x = 55

What is 35% of 160?

Call the unknown x. Translate the sentence, replacing "is" by "=" and the unknown by x. is/of = % / 100 x/160 = 35/100 => 100x = 35(160) => 100x = 5,600 => x = 56

Which of the following numbers satisfies the inequality 2/3 < 1/x < 3/4? A. 3/5 B. 7/10 C. 13/10 D. 10/7 E. 5/3

Convert the fractions into decimals. Then the inequality becomes 0.666... < 1/x < 0.75. There are infinitely many values of 1/x that satisfy this inequality, but 1/x = 0.7 is the obvious choice. Then 1/x = 7/10 => x = 10/7 D. 10/7

On a test consisting of 80 questions, Marie answered 75% of the first 60 questions correctly. What percent of the other 20 questions did she need to answer correctly for her grade on the entire exam to be 80%. A. 85% B. 87.5% C. 90% D. 95% E. 100%

D. 95%. To earn an 80% on the entire exam, Marie needs to answer a total of 64 questions correctly (80% of 80). So far, she has answered 45 questions correctly (75% of 60). Therefore, on the last 20 questions, she needs 64 - 46 = 19 correct answers, and 19/20 = 95%

At Sally's sale shop everything is sold for 20% less than the price marked. If Sally buys sweaters for $80, what price should she mark them if she wants to make a 20% profit on her cost? A. $80 B. $96 C. $100 D. $120 E. $125

D. Since 20% of 80 is 16, Sally wants to get $80 + $16 = $96 for each sweater she sells. Then if x is the marked price: $96 = x - 0.20x = 0.8x => x = ($96)/(0.8) =$120

Which of the following is not an irrational number? A. π B. √(π²) C. (π³)/(π²) D. √(π²) / π E. π⁰/π

D. √(π²) / π => π/π => 1 A, B, and C all all equal to π which is irrational. E is 1/π, the reciprocal of an irrational is still an irrational.

If from 1980 to 1990 the population of a town increased from 12,000 to 15,000 and from 1990 to 2000 the population increased by the same percent, then what was the percent increase in population from 1980 to 2000?

From 1980 to 1990 the population increased by 25%. percent increase: (the actual increase)/(the original amount) x 100% 3,000/12,000 x 100% = 25% If from 1990 to 2000 the population again increased by 25%, it increased by (0.25)(15,000) = 3,750 So, the population in 2000 was 18,750. Therefore, from 1980 to 2000, the population increased by 18,750 -12,000 = 6,750. The percent increase was 6,750/12,000 x 100% = 56.25%

rational number

Numbers that can be expressed as fractions

Starting on January 1, 1990, the population of Centerville increased by 5% every year. if Centerville's population was 10,000 on January 1, 1990, what was its population on January 1, 1992?

Since the population on January 1, 1990 was 10,000, to find the population on year later on January 1, 1991, multiply it by 1.05: (1 + r%) (1+5%)(10,000) => (1+0.05)(10,000) => 10,500 To get the population on January 1, 1992, again multiply by 1.05. (1.05)(10,500) = 11,025. Note that (1.05)(10,500) = (1.05)[ (1.05)(10,000) ] = (1.05)²(10,000). This process can be continued for any number of years.

Evaluate 1 2/3 x 3 1/4

THESE ARE MIXED NUMBERS (1+2/3)(3+1/4) = 65/12 or 5 5/12 or 5.416666

Simplify (1+1/6) / (2-3/4)

Version 1: Multiply each term by 12, the LCM of 6 and 4 (1+1/6) / (2-3/4) = > (12 + 2) / (24 - 9) => 14/15 Version 2: Simplify numerator and denominator then divide (1+1/6) / (2-3/4) => (7/6) / (5/4) => (7/6) x (4/5) => 14/15

To multiply a fraction by any other number, write that number as ______________.

a fraction whose denominator is 1

If ad < bc, then _________.

a/b < c/d

If ad = bc, then _________.

a/b = c/d

If ad > bc, then _________.

a/b > c/d

a/b ÷ c/d = ?

a/b ÷ c/d = (a/b)(d/c) => ad/bc

To add or subtract fractions with the same denominator, ________________________________.

add or subtract the numerators and keep the denominator.

A __________ is a fraction that has a fraction in its numerator,denominator, or both.

complex fraction

To determine if two fractions are equivalent (have the same value) or if one is greater than the other, __________.

cross multiply

A __________ is a number such as 3 1/2 that consists of an integer followed by a fraction. It is an abbreviation for the sum of the integer and the fraction; so, 3 1/2 = 3+1/2

mixed number

To convert a percent to a decimal, drop the % symbol and __________________, adding 0's if necessary.

move the decimal point two places to the left

To multiply two fractions, __________________________.

multiply their numerators and multiply their denominators.

Does the decimal repeat? 3/8

no. 3/8 = 0.375

Does the decimal repeat? 7/10

no. 7/10 = 0.7

irrational numbers

numbers that cannot be expressed as fractions

The word __________ means hundredth. The symbol % is used to express the word. For example, 23 __________ means 23 hundredths and can be written with a % symbol, as a fraction, or as a decimal.

percent

From 1980 to 1990, the population of a town increased from 12,000 to 15,000. Since the actual increase in the population was 3,000, the percent increase in the population was

percent increase: (the actual increase)/(the original amount) x 100% 3,000/12,000 x 100% = 25%

4, 0.7, (that line thing above it→)0.22222, 20%, and 3/11 are examples of ___________________ numbers.

rational numbers

The __________ of any nonzero number x, is the number 1/x. The __________ of the fraction a/b is b/a.

reciprocal

To divide any number by a fraction, multiply that number by the __________ of the fraction.

reciprocal

If a fraction is written in lowest terms and if the denominator has any prime factor other than 2 and 5, the decimal __________.

repeats

To add or subtract fractions with different denominators, first ______________________________.

rewrite the fractions with the same denominator

If a fraction is written in lowest terms and if the only prime factors of the denominator are 2 and 5, the decimal __________.

terminates

To convert a percent to a fraction, drop the % symbol, ______________________, and reduce.

write the number over 100

Does the decimal repeat? 17/12

yes. 17/12 = (that line thing above it→) 1.21666. The denominator has a prime other than two or five.

Does the decimal repeat? 5/6

yes. 5/6 = (that line thing above it→) 0.8333. The denominator has a prime other than two or 5.

Which of these numbers are irrational? π, ³√8, √8

π and √8 are irrational. ³√8 is NOT irrational because it equal to 2 (2^3 = 8) and therefore can be expressed as a fraction.

Evaluate √2 ÷ 2/3

√2 ÷ 2/3 => √2/1 ÷ 3/2 => 3√2/2