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The equivalent rational number of 17/9, whose numerator is 136 is _________
Explanation To find the equivalent rational number of 17/9 with a numerator of 136, we need to find a common factor that can be multiplied to both the numerator and denominator of 17/9 to obtain 136 as the numerator. By dividing 136 by 17, we find that the common factor is 8. Therefore, multiplying both the numerator and denominator of 17/9 by 8 gives us the equivalent rational number of 136/72.
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2.
(534 x 991)^{-1}= (534)^{-1 }x _________
Explanation The expression (534 x 991)-1 can be simplified by multiplying 534 and 991 first, which equals 529,794. Subtracting 1 from this result gives us 529,793. On the other side of the equation, (991)-1 simplifies to 990. Therefore, the missing value that completes the equation is 990.
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3.
The rational number 9.99 in the form of p/q is _________
Explanation The rational number 9.99 can be expressed as the fraction 999/100. This is because the decimal 9.99 can be written as 9 + 0.99, and 0.99 can be expressed as 99/100. Therefore, 9.99 can be written as 9 + 99/100, which simplifies to 999/100.
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4.
^{1/15 x [27/31 +32/37] = [1/15 x 27/31] + __________. name the property}
Explanation The given equation involves the distributive property, which states that when multiplying a number by a sum, you can multiply each term in the sum separately and then add the products. In this case, the equation can be rewritten as [1/15 x 27/31] + [1/15 x 32/37]. Therefore, the correct answer is using the distributive property.
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5.
The rational numbers 4/17 and -4/17 are on the _____ sides of zero on the number line.
A.
Opposite
B.
Same
Correct Answer A. Opposite
Explanation The rational numbers 4/17 and -4/17 are on the opposite sides of zero on the number line because one is positive (4/17) and the other is negative (-4/17).
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6.
4/7 is ____________ than - 4/7
A.
Greater
B.
Smaller
Correct Answer B. Smaller
Explanation When comparing two fractions, the larger fraction is the one with the greater numerator or the smaller denominator. In this case, 4/7 has a greater numerator than -4/7, making it the larger fraction. Therefore, -4/7 is smaller than 4/7.
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7.
There are _______ rational numbers between any two rational numbers
Correct Answer infinite
Explanation The statement "There are infinite rational numbers between any two rational numbers" is a fundamental property of rational numbers. This is because rational numbers can be expressed as fractions, where the numerator and denominator can take on any integer value. Since there are infinitely many integers, there are infinitely many possible fractions between any two rational numbers. Therefore, the correct answer is infinite.
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8.
The additive inverse of a positive rational is always a ____________ rational number
Correct Answer negative
Explanation The additive inverse of a positive rational number is always a negative rational number. This is because the additive inverse of any number is the number that when added to the original number, results in zero. Since the original number is positive, its additive inverse must be negative in order to cancel out its positive value and produce zero. Therefore, the correct answer is negative.
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9.
The reciprocal of - 15/17 is ______
Correct Answer - 17/15
Explanation The reciprocal of a fraction is obtained by flipping the numerator and denominator. In this case, the reciprocal of -15/17 would be -17/15.
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10.
Rational number - 3/5 lies between consecutive integers (-1) and __________
Correct Answer zero
Explanation The rational number 3/5 lies between consecutive integers -1 and 0.
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11.
1. If a/b is a rational number, then "b" can be any whole number
A.
True
B.
False
Correct Answer B. False
Explanation If a/b is a rational number, then "b" cannot be any whole number. A rational number is a number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. However, the denominator cannot be any whole number because if it is zero, the fraction becomes undefined. Therefore, the statement "b can be any whole number" is false.
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12.
- 5/10 lies between - 1/2 and 1
A.
True
B.
False
Correct Answer B. False
Explanation The statement 5/10 lies between -1/2 and 1 is false. 5/10 is equal to 1/2, which is greater than -1/2 but less than 1. Therefore, the statement is incorrect.
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13.
If p ≠ 0, the multiplicative inverse of p/q is q/p?
A.
True
B.
False
Correct Answer A. True
Explanation The statement is true because the multiplicative inverse of a fraction p/q is obtained by interchanging the numerator and denominator, resulting in q/p. Since the condition given is p â‰ 0, it is valid to say that the multiplicative inverse of p/q is indeed q/p.
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14.
The negative of the negative of any rational number is the number itself
A.
True
B.
False
Correct Answer A. True
Explanation The statement is true because when we negate a rational number, we change its sign. Negating it again brings it back to its original sign, resulting in the number itself. This property holds true for all rational numbers.
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15.
The negative of 0 does not exist
A.
True
B.
False
Correct Answer A. True
Explanation The statement is true because the negative of a number is the opposite of that number. However, in the case of 0, there is no number that can be its opposite because any number multiplied by 0 is always 0. Therefore, the negative of 0 does not exist.
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16.
For any rational number "a" and "b", "a - b" = "b - a"
A.
True
B.
False
Correct Answer B. False
Explanation The given statement "a - b = b - a" is not true for any rational numbers a and b. The subtraction operation is not commutative, meaning that changing the order of the numbers being subtracted will yield different results. Therefore, the correct answer is False.
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17.
For every rational number x, y, and z. x + (y x z) = (x + y) x (x + z)
A.
True
B.
False
Correct Answer A. True
Explanation The given equation states that for any rational numbers x, y, and z, the sum of x and the product of y and z is equal to the product of the sum of x and y, and the sum of x and z. This equation is true for all rational numbers because it follows the distributive property of multiplication over addition.
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18.
1 is the only number, which is own reciprocal
A.
True
B.
False
Correct Answer B. False
Explanation The statement "1 is the only number, which is own reciprocal" is incorrect. The reciprocal of a number is obtained by dividing 1 by that number. In the case of 1, its reciprocal is also 1 because 1 divided by 1 is 1. Therefore, 1 is not the only number that is its own reciprocal. Other numbers like -1, 2, and -2 also have the property of being their own reciprocals.
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19.
-1 is not the reciprocal of any rational number
A.
True
B.
False
Correct Answer B. False
Explanation The statement "1 is not the reciprocal of any rational number" is false. The reciprocal of any rational number can be found by interchanging the numerator and denominator. For example, the reciprocal of 2/3 is 3/2. Therefore, 1 can be considered the reciprocal of itself, as 1/1 is equal to 1.
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20.
If X and Y are negative rational numbers, then so is ( X + Y)
A.
True
B.
False
Correct Answer A. True
Explanation If X and Y are negative rational numbers, it means that they can be expressed as a fraction where the numerator and denominator are both negative integers. When we add two negative rational numbers, the negative signs of the numerator and denominator cancel out, resulting in a positive rational number. Therefore, (X + Y) will also be a negative rational number. Hence, the statement is true.
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21.
The reciprocal of X^{-1 }is 1/X
A.
True
B.
False
Correct Answer B. False
Explanation The reciprocal of X-1 is actually 1/(X-1), not 1/X. This is because the reciprocal of a number is obtained by flipping the number over, so the reciprocal of X-1 would be 1 divided by (X-1), not 1 divided by X. Therefore, the statement is false.
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22.
Zero is the smallest rational number
A.
True
B.
False
Correct Answer B. False
Explanation The statement "Zero is the smallest rational number" is false. A rational number is a number that can be expressed as a fraction, where the numerator and denominator are both integers. Zero can be expressed as 0/1, which is a fraction, but it is not the smallest rational number. The smallest rational number is actually 1/âˆž (one divided by infinity), as it is the closest to zero without actually being zero.
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23.
X + Y/ 2 is a rational number
A.
Between X and Y
B.
Less than X and Y both
C.
Greater than X and Y both
D.
Less than X but greater than Y
Correct Answer A. Between X and Y
Explanation The expression X + Y/2 represents the average of X and Y. Since it is the average, it will always be between X and Y. Therefore, the correct answer is "between X and Y".
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24.
The reciprocal of zero is
Correct Answer not defined
Explanation The reciprocal of a number is obtained by dividing 1 by that number. However, dividing any number by zero is undefined in mathematics. This is because division by zero violates the fundamental properties of arithmetic and leads to contradictions. Therefore, the reciprocal of zero is not defined.
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25.
Zero is
A.
The identity for addition of rational numbers
B.
The identity for subtraction of rational numbers
C.
The identity for multiplication of rational numbers
D.
The identity for division of rational numbers
Correct Answer A. The identity for addition of rational numbers
Explanation Zero is the identity for addition of rational numbers because when we add any rational number to zero, the result is always the same rational number. In other words, zero does not change the value of any rational number when added to it. This property is known as the identity property of addition, and zero is the unique element that satisfies this property for rational numbers.
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26.
One (1) is
A.
The identity for addition of rational numbers
B.
The identity for subtraction of rational numbers
C.
The identity for multiplication of rational numbers
D.
The identity for division of rational numbers
Correct Answer C. The identity for multiplication of rational numbers
Explanation The identity for multiplication of rational numbers is 1. This means that when any rational number is multiplied by 1, the result is always equal to the original rational number. In other words, 1 acts as a neutral element for multiplication in the set of rational numbers.
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27.
Multiplicative inverse of a negative rational number is
A.
0
B.
-1
C.
A negative rational number
D.
A positive rational number
Correct Answer C. A negative rational number
Explanation The multiplicative inverse of a number is the reciprocal of that number. In the case of a negative rational number, the reciprocal will also be negative, as dividing a negative number by another negative number results in a positive number. Therefore, the multiplicative inverse of a negative rational number is a negative rational number.
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28.
To get the product - 4/5, we should multiply 10/11 by
Correct Answer -22/25
Explanation To get the product -4/5, we need to multiply 10/11 by -22/25. Multiplying these fractions gives us (-22/25) * (10/11) = -220/275. Simplifying this fraction gives us -4/5, which is the desired product.
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29.
By what numbers should we multiply -15/20, so that the product may be -5/7?
Correct Answer 20/21
Explanation To find the numbers we should multiply -15/20 by in order to get a product of -5/7, we need to divide -5/7 by -15/20. Dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. So, we can multiply -5/7 by the reciprocal of -15/20, which is 20/15. Simplifying this product gives us -100/105, which can be simplified further to -20/21. Therefore, the numbers we should multiply -15/20 by to get a product of -5/7 is 20/21.
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30.
A train travels 1445/2 km in 17/2 hrs. Find the speed of the train in km/h?
Correct Answer 85
Explanation The speed of the train can be found by dividing the distance traveled by the time taken. In this case, the train travels 1445/2 km in 17/2 hours. To find the speed, we divide 1445/2 by 17/2, which simplifies to 85. Therefore, the speed of the train is 85 km/h.