Grade 9 Math: Number Sense

Rounding to the nearest tenth, values from $4.95$ up to but not including $5.05$ round to $5.0$. Include $4.95$; exclude $5.05$ since it rounds to $5.1$.

There are infinitely many rationals between two decimals. $0.305$ lies between $0.3$ and $0.31$. $0.31$ and $0.300$ are not strictly between, and $\sqrt{2}$ is irrational.

Compute the unit rate: $\dfrac{450}{30}=15\,\text{km/L}$. Divide distance by litres.

Add 13%: $49.99\times1.13=56.4887$. Rounding to the nearest cent gives $\$56.49$.

$\sqrt{5}\approx2.236$ is between 2 and 3 and is irrational. $\pi\approx3.141$ is greater than 3, $\sqrt{9}=3$, and $2.5$ is rational.

$2\frac{1}{3}=\frac{7}{3}$. Multiply: $\frac{7}{3}\cdot\frac{3}{5}=\frac{21}{15}$, then simplify by $3$ to get $\frac{7}{5}$.

A repeating decimal represents a fraction, so it is rational. It is not an integer or a natural number.

$3.2\times50{,}000=160{,}000$ cm $=1{,}600$ m $=1.6$ km. So the actual distance is 1.6 km.

$2.5\%=0.025=\frac{25}{1000}$. Divide numerator and denominator by 25 to get $\frac{1}{40}$.

Square first: $(-3)^2=9$. Then multiply: $-4\cdot(-2)=8$. Add: $9+8+5=22$. The double negative in the product makes it positive.

LCM$(8,12)=24$. Convert: $\dfrac{3}{8}=\dfrac{9}{24}$ and $\dfrac{5}{12}=\dfrac{10}{24}$. Sum: $\dfrac{9}{24}+\dfrac{10}{24}=\dfrac{19}{24}$.

A repeating decimal is a fraction: $0.\overline{27}=\dfrac{27}{99}=\dfrac{3}{11}$. It is rational, and it is not an integer.

$\sqrt{2}$ cannot be expressed as a fraction, so it is irrational. The others are rational: a fraction, a repeating decimal ($1/3$), and an integer.

Rational numbers are dense: between any two distinct real numbers there are infinitely many rationals. For instance, many fractions like $-4.9,-4.95,-4.99$, etc.

Dividing by a fraction means multiply by its reciprocal: $\frac{-3}{5}\times\frac{10}{9}=\frac{-30}{45}=-\frac{2}{3}$ after simplifying.

Dividing by a fraction means multiply by its reciprocal: $\frac{5}{8}\cdot\frac{3}{10}=\frac{15}{80}$, which simplifies to $\frac{3}{16}$.

$2\frac{1}{2}=2.5$. Then $2.5+0.75=3.25$. As a mixed number, this is $3\tfrac{1}{4}\,\text{L}$.

At $1:250{,}000$, $1\,\text{cm}$ represents $250{,}000\,\text{cm}=2.5\,\text{km}$. For $3\,\text{cm}$, $3\times2.5=7.5\,\text{km}$.

$\sqrt{2}$ cannot be written as a ratio of integers. The others are rational: $\sqrt{81}=9$, $0.125=\frac{1}{8}$, and $\frac{22}{7}$ is a fraction.

Use a proportion: $8/100=x/250$. Solve to get $x=8\times2.5=20$. The car needs 20 L.

Total $=64\times1.13=72.32$. HST adds $13\%$ of the price to the subtotal.

On the number line, numbers farther right are greater. $2$ is to the right of $0$, $-3$, and $-7$, so it is greatest.

Compute $(-3)^2=9$. Then $-4\cdot(-2)=+8$. The absolute value $|{-7}|=7$. Add them: $9+8+7=24$.

To be strictly between, it must be greater than $1.2$ and less than $1.3$. $1.21$ meets this; $1.20$ and $1.30$ are endpoints, and $1.35$ is larger.

$\sqrt{9}=3$, so $-\sqrt{9}=-3$. That is an integer. Integers are a subset of rational and real numbers, but negative integers are not natural or whole.

Use a common denominator of $8$. Convert $\frac{3}{4}$ to $\frac{6}{8}$. Then $\frac{6}{8}-\frac{5}{8}=\frac{1}{8}$.

First evaluate inside absolute values: $| -7 |=7$ and $3-9=-6$ so $|3-9|=6$. Then $7-6=1$.

Average speed is distance divided by time: $\frac{600}{7.5}=80$. Include units: km/h.

Change equals final minus initial: $3-(-12)=15$. The positive result means an increase of $15^\circ\text{C}$.

Start at $-4$. Subtract $6$ to get $-10$. Add $9$ to get $-1$. Therefore the noon temperature is $-1^\circ\text{C}$.

Use a proportion: $8.2$ L per 100 km means $8.2\times\frac{350}{100}=28.7$ L.

Apply exponents before the negative sign: $-3^2=-(3^2)=-9$. Then add and subtract: $-9+5=-4$, and $-4-(-4)=-4+4=0$.

Start at $-8$. Add $5$ to get $-3$. Subtract $12$ to get $-15$. Keep the temperature unit.

Common denominator 12: $\frac{3}{4}=\frac{9}{12}$ and $\frac{5}{6}=\frac{10}{12}$. Subtract to get $\frac{9}{12}-\frac{10}{12}=-\frac{1}{12}$.

Evaluate the bracket first: $4-(-7)=11$. Then $-18-11=-29$.

Any integer $n$ can be written as $\dfrac{n}{1}$, so integers are rational. The other statements contradict definitions.

$3.15$ is greater than $3.1$ and less than $3.2$. $3.05<3.1$, $3.25>3.2$, and $3.20=3.2$ so it is not strictly between.

$\sqrt{20}=2\sqrt{5}$ is irrational. Negating it keeps it irrational, so it is not rational, an integer, or whole.

Divide numerator and denominator by $6$: $\dfrac{18}{-24}=\dfrac{3}{-4}=-\dfrac{3}{4}$. A single negative makes the result negative.

$-5$ is a negative whole number, so it is an integer. Integers are a subset of rational and real numbers. It is not whole or natural.

Each term gets closer to $1$: the difference is $0.1, 0.01, 0.001,\dots$. The limit (approach value) is $1$.

$\frac{4}{5}=0.8$ and $\frac{5}{6}\approx0.8333$. $\frac{41}{50}=0.82$ lies between them; the other options are outside that interval.

At $1:50{,}000$, 1 cm represents 50,000 cm $=0.5$ km. So 3 cm represents $3\times0.5=1.5$ km.

$\sqrt{49}=7$. Seven is a natural number. Natural numbers are a subset of integers and rationals, so it is the smallest appropriate set.

Since $49<50<64$, $7<\sqrt{50}<8$. More precisely, $\sqrt{50}\approx7.071$. Rounding to the nearest hundredth gives $7.07$.