Quiz 1
Arithmetic fundamentals practice
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In a certain city, the ratio of apartments to houses is $$9:5$$. If there are $$160$$ more apartments than houses, what is the total number of residences in the city?⌄
Let apartments be $$9x$$ and houses be $$5x$$. Difference: $$9x-5x=4x$$ $$4x=160$$ $$x=40$$ Total residences: $$9x+5x=14x$$ $$14x=560$$ Final answer: $$560$$
In a school, every student is either in math club or science club. The math club members make up $$\frac{7}{15}$$ of the students. Compare Column A and Column B.Column A: The ratio of math club students to science club students Column B: The fraction of students who are in the science club⌄
Math fraction: $$\frac{7}{15}$$ Science fraction: $$1-\frac{7}{15}=\frac{8}{15}$$ Column A: $$\frac{7}{15}:\frac{8}{15}=\frac{7}{8}$$ Column B: $$\frac{8}{15}$$ Compare: $$\frac{7}{8}=0.875$$ $$\frac{8}{15}\approx0.533$$ So Column A is greater. Final answer: $$\text{Column A is greater}$$
After receiving a $$20\%$$ discount, a customer paid $$240$$ dollars for a jacket. What was the original price before the discount?⌄
Let original price be $$x$$. After $$20\%$$ discount: $$0.8x=240$$ Solve: $$x=\frac{240}{0.8}$$ $$x=300$$ Final answer: $$300$$
What fraction is $$40\%$$ of $$60$$ of $$25\%$$ of $$120$$?⌄
Compute each part. $$40\%\text{ of }60=0.4\cdot60=24$$ $$25\%\text{ of }120=0.25\cdot120=30$$ Form the fraction. $$\frac{24}{30}$$ Simplify. $$\frac{4}{5}$$ Final answer: $$\frac{4}{5}$$
Four coworkers divide $$96{,}000$$ dollars in the ratio $$2:3:4:7$$. Compare Column A and Column B.Column A: The difference between the largest share and the smallest share Column B: $30,000⌄
Total ratio: $$2+3+4+7=16$$ Each part: $$\frac{96000}{16}=6000$$ Largest share: $$7\cdot6000=42000$$ Smallest share: $$2\cdot6000=12000$$ Difference: $$42000-12000=30000$$ Compare: $$30000=30000$$ Final answer: $$\text{Equal}$$
Column A: The number of distinct prime factors of $$90$$ Column B: The number of distinct prime factors of $$126$$⌄
Prime factorization. $$90=2\cdot3^2\cdot5$$ Distinct primes: $$2,3,5$$ Count: $$3$$ $$126=2\cdot3^2\cdot7$$ Distinct primes: $$2,3,7$$ Count: $$3$$ Final answer: $$\text{Equal}$$
The greatest common factor of $$54$$ and $$81$$ is:⌄
Prime factorization. $$54=2\cdot3^3$$ $$81=3^4$$ Common factor: $$3^3$$ $$3^3=27$$ Final answer: $$27$$
If $$k$$ is an odd integer, which expression must be an even integer?⌄
Let $$k$$ be odd. $$k=2n+1$$ Check each expression. $$k^2+1=(2n+1)^2+1=4n^2+4n+2$$ Even. $$4k+3=4(2n+1)+3=8n+7$$ Odd. $$2k=2(2n+1)=4n+2$$ Even. $$\frac{5k}{2}$$ not always integer. $$k^3=(2n+1)^3$$ Odd. Expressions that must be even: $$k^2+1,\ 2k$$ Final answer: $$\text{k}^2+1\text{ and }2k$$