Rational Numbers - Class 7 - Mathematics
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Examples - Rational Numbers | NCERT | Mathematics | Class 7
$\frac{-45}{30}$ to the standard form.
To convert $\frac{-45}{30}$ to standard form, we simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by this GCD.
The GCD of $45$ and $30$ is $15$, so we divide both the numerator and the denominator by $15$:
$$\frac{-45}{30} = \frac{-45 \div 15}{30 \div 15} = \frac{-3}{2}$$
Therefore, the fraction $\frac{-45}{30}$ in standard form is $\frac{-3}{2}$.
Reduce to standard form: show steps
(i) $\frac{36}{-24}$
(ii) $\frac{-3}{-15}$
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Sign up nowDo $\frac{4}{-9}$ and $\frac{-16}{36}$ represent the same rational number?
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Sign up nowList three rational numbers between-2 and-1.
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Sign up nowWrite four more numbers in the following pattern:
$\frac{-1}{3}, \frac{-2}{6}, \frac{-3}{9}, \frac{-4}{12}, \ldots$
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Sign up nowSatpal walks $\frac{2}{3} \mathrm{~km}$ from a place P, towards east and then from there
$1 \frac{5}{7} \mathrm{~km} \text { towards west. Where will he be now from } \mathrm{P} \text { ? }$
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Sign up nowExercise 8.1 - Rational Numbers | NCERT | Mathematics | Class 7
List five rational numbers between:
(i) -1 and 0
(ii) -2 and -1
(iii) $\frac{-4}{5}$ and $\frac{-2}{3}$
(iv) $-\frac{1}{2}$ and $\frac{2}{3}$
(i) Between -1 and 0
To find rational numbers between -1 and 0, think of splitting the interval into equal parts or consider common fractions within the range. A straightforward approach is to use fractions with a denominator of 5 or 10.
-4/5
-3/5
-2/5
-1/5
-1/10
(ii) Between -2 and -1
Similar to the first case, divide the interval into equal parts or use common fractions.
-7/4
-3/2
-5/4
-6/5
-11/10
(iii) Between $\frac{-4}{5}$ and $\frac{-2}{3}$
Finding common ground for the denominators (here, consider multiples of both 5 and 3) can help. Denominators of 15 can be practical.
$-\frac{11}{15}$
$-\frac{10}{15} = -\frac{2}{3}$ (include to illustrate the border, we'll seek another)
$-\frac{9}{15} = -\frac{3}{5}$
$-\frac{8}{15}$
Another can be added between $-\frac{3}{5}$ and $-\frac{4}{5}$, such as $-\frac{7}{15}$.
(iv) Between $-\frac{1}{2}$ and $\frac{2}{3}$
For this range, selecting a denominator that is a multiple of both 2 and 3, such as 12, would allow a broader range of rational numbers.
$-\frac{5}{12}$
$-\frac{4}{12} = -\frac{1}{3}$
$-\frac{2}{12} = -\frac{1}{6}$
$-\frac{1}{12}$
$\frac{1}{12}$
Each step involves selecting fractions with denominators that allow the fractions to fall within the given intervals. Adjustments can be made to the choice of fractions based on the desired accuracy or density of numbers within the interval.
Write four more rational numbers in each of the following patterns:
(i) $\frac{-3}{5}, \frac{-6}{10}, \frac{-9}{15}, \frac{-12}{20}, \ldots$.
(ii) $\frac{-1}{4}, \frac{-2}{8}, \frac{-3}{12}, \ldots$.
(iii) $\frac{-1}{6}, \frac{2}{-12}, \frac{3}{-18}, \frac{4}{-24}, \ldots .$.
(iv) $\frac{-2}{3}, \frac{2}{-3}, \frac{4}{-6}, \frac{6}{-9}, \ldots$.
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Sign up nowGive four rational numbers equivalent to:
(i) $\frac{-2}{7}$
(ii) $\frac{5}{-3}$
(iii) $\frac{4}{9}$
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Sign up nowDraw the number line and represent the following rational numbers on it:
(i) $\frac{3}{4}$
(ii) $\frac{-5}{8}$
(iii) $\frac{-7}{4}$
(iv) $\frac{7}{8}$
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Sign up nowThe points $\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}, \mathrm{T}, \mathrm{U}, \mathrm{A}$ and $\mathrm{B}$ on the number line are such that, $\mathrm{TR}=\mathrm{RS}=\mathrm{SU}$ and $\mathrm{AP}=\mathrm{PQ}=\mathrm{QB}$. Name the rational numbers represented by $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$. with steps
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Sign up nowWhich of the following pairs represent the same rational number? with steps
(i) $\frac{-7}{21}$ and $\frac{3}{9}$
(ii) $\frac{-16}{20}$ and $\frac{20}{-25}$
(iii) $\frac{-2}{-3}$ and $\frac{2}{3}$
(iv) $\frac{-3}{5}$ and $\frac{-12}{20}$
(v) $\frac{8}{-5}$ and $\frac{-24}{15}$
(vi) $\frac{1}{3}$ and $\frac{-1}{9}$
(vii) $\frac{-5}{-9}$ and $\frac{5}{-9}$
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Sign up nowRewrite the following rational numbers in the simplest form:
(i) $\frac{-8}{6}$
(ii) $\frac{25}{45}$
(iii) $\frac{-44}{72}$
(iv) $\frac{-8}{10}$
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Sign up nowFill in the boxes with the correct symbol out of $>,<$, and $=$. with steps
(i) $\frac{-5}{7} \square \frac{2}{3}$
(ii) $\frac{-4}{5} \square \frac{-5}{7}$
(iii) $\frac{-7}{8} \square \frac{14}{-16}$
(iv) $\frac{-8}{5} \square \frac{-7}{4}$
(v) $\frac{1}{-3} \square \frac{-1}{4}$
(vi) $\frac{5}{-11} \square \frac{-5}{11}$
(vii) $0 \square \frac{-7}{6}$
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Sign up nowWhich is greater in each of the following: with steps
(i) $\frac{2}{3}, \frac{5}{2}$
(ii) $\frac{-5}{6}, \frac{-4}{3}$
(iii) $\frac{-3}{4}, \frac{2}{-3}$
(iv) $\frac{-1}{4}, \frac{1}{4}$
(v) $-3 \frac{2}{7},-3 \frac{4}{5}$
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Sign up nowWrite the following rational numbers in ascending order:
(i) $\frac{-3}{5}, \frac{-2}{5}, \frac{-1}{5}$
(ii) $\frac{-1}{3}, \frac{-2}{9}, \frac{-4}{3}$
(iii) $\frac{-3}{7}, \frac{-3}{2}, \frac{-3}{4}$
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Sign up nowExercise 8.2 - Rational Numbers | NCERT | Mathematics | Class 7
Find the sum:
(i) $\frac{5}{4}+\left(\frac{-11}{4}\right)$
(ii) $\frac{5}{3}+\frac{3}{5}$
(iii) $\frac{-9}{10}+\frac{22}{15}$
(iv) $\frac{-3}{-11}+\frac{5}{9}$
(v) $\frac{-8}{19}+\frac{(-2)}{57}$
(vi) $\frac{-2}{3}+0$
(vii) $-2 \frac{1}{3}+4 \frac{3}{5}$
To find the sum of each pair of fractions and mixed numbers, follow these steps:
(i) $\frac{5}{4}+\left(\frac{-11}{4}\right)$
Step 1: Since the denominators are the same, simply add the numerators: $5 + (-11) = -6$.
Step 2: Keep the same denominator: $\frac{-6}{4}$.
Step 3: Simplify if possible: $\frac{-6}{4} = \frac{-3}{2}$.
(ii) $\frac{5}{3}+\frac{3}{5}$
Step 1: Find a common denominator, which is $15$ in this case.
Step 2: Convert the fractions: $\frac{5}{3} = \frac{25}{15}$ and $\frac{3}{5} = \frac{9}{15}$.
Step 3: Add the converted fractions: $\frac{25}{15} + \frac{9}{15} = \frac{34}{15}$.
(iii) $\frac{-9}{10}+\frac{22}{15}$
Step 1: Find a common denominator, which is $30$.
Step 2: Convert the fractions: $\frac{-9}{10} = \frac{-27}{30}$ and $\frac{22}{15} = \frac{44}{30}$.
Step 3: Add the converted fractions: $\frac{-27}{30} + \frac{44}{30} = \frac{17}{30}$.
(iv) $\frac{-3}{-11}+\frac{5}{9}$
Step 1: Simplify the first fraction: $\frac{-3}{-11} = \frac{3}{11}$.
Step 2: Since the denominators are different and there's no simple common denominator, it's easiest to find their product: $99$.
Step 3: Convert the fractions: $\frac{3}{11} = \frac{27}{99}$ and $\frac{5}{9} = \frac{55}{99}$.
Step 4: Add the converted fractions: $\frac{27}{99} + \frac{55}{99} = \frac{82}{99}$.
(v) $\frac{-8}{19}+\frac{(-2)}{57}$
Step 1: Find a common denominator, which is $57 \times 19 = 1083$.
Step 2: Convert the fractions: $\frac{-8}{19} = \frac{-456}{1083}$ and $\frac{-2}{57} = \frac{-38}{1083}$.
Step 3: Add the converted fractions: $\frac{-456}{1083} + \frac{-38}{1083} = \frac{-494}{1083}$.
Step 4: Simplify if possible.
(vi) $\frac{-2}{3}+0$
Step 1: Adding zero to a number does not change its value: $\frac{-2}{3}$.
(vii) $-2 \frac{1}{3}+4 \frac{3}{5}$
Step 1: Convert the mixed numbers to improper fractions: $-2 \frac{1}{3} = \frac{-7}{3}$ and $4 \frac{3}{5} = \frac{23}{5}$.
Step 2: Find a common denominator, which is $15$.
Step 3: Convert the fractions: $\frac{-7}{3} = \frac{-35}{15}$ and $\frac{23}{5} = \frac{69}{15}$.
Step 4: Add the converted fractions: $\frac{-35}{15} + \frac{69}{15} = \frac{34}{15}$.
Please note, for simplification of the fractions and finding common denominators, it's done manually here. In actual computation, simplification steps might vary depending on the fraction's nature.
Find
(i) $\frac{7}{24}-\frac{17}{36}$
(ii) $\frac{5}{63}-\left(\frac{-6}{21}\right)$
(iii) $\frac{-6}{13}-\left(\frac{-7}{15}\right)$
(iv) $\frac{-3}{8}-\frac{7}{11}$
(v) $-2 \frac{1}{9}-6$
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Sign up nowFind the product:
(i) $\frac{9}{2} \times\left(\frac{-7}{4}\right)$
(ii) $\frac{3}{10} \times(-9)$
(iii) $\frac{-6}{5} \times \frac{9}{11}$
(iv) $\frac{3}{7} \times\left(\frac{-2}{5}\right)$
(v) $\frac{3}{11} \times \frac{2}{5}$
(vi) $\frac{3}{-5} \times \frac{-5}{3}$
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Sign up nowFind the value of:
(i) $(-4) \div \frac{2}{3}$
(ii) $\frac{-3}{5} \div 2$
(iii) $\frac{-4}{5} \div(-3)$
(iv) $\frac{-1}{8} \div \frac{3}{4}$
(v) $\frac{-2}{13} \div \frac{1}{7}$
(vi) $\frac{-7}{12} \div\left(\frac{-2}{13}\right)$
(vii) $\frac{3}{13} \div\left(\frac{-4}{65}\right)$
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Sign up nowExtra Questions - Rational Numbers | NCERT | Mathematics | Class 7
How is $\frac{12}{25}$ represented as a decimal?
(A) 4.8
(B) 48
(C) 0.048
(D) 0.48
Correct option: $\mathbf{D}$ (0.48)
To convert $\frac{12}{25}$ to its decimal form, one approach is to manipulate the denominator so that it becomes a power of 10, such as 100, which makes the decimal conversion straightforward.
We begin by finding an equivalent fraction with a denominator of 100: $$ \frac{12 \times 4}{25 \times 4} = \frac{48}{100} $$
With the fraction in this format, converting to a decimal is simple. Since the denominator is 100, the numerator—48—represents how many hundredths there are, placing the decimal two places to the left: $$ 0.48 $$
Hence, $\frac{12}{25}$ as a decimal is $0.48$, corresponding to option (D).
Between the numbers $\frac{15}{20}$ and $\frac{35}{40}$, the greater number is:
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