Grade 8 Math Part 1 Rational Numbers Unit 1

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).

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 ≠ 0, it is valid to say that the multiplicative inverse of p/q is indeed q/p.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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/∞ (one divided by infinity), as it is the closest to zero without actually being zero.

An integer is considered a rational number because it can be expressed as a fraction where the denominator is 1. For example, the integer 5 can be written as 51\frac{5}{1}15​, which qualifies it as a rational number. Rational numbers are defined as any number that can be expressed as a ratio of two integers. Therefore, all integers are rational numbers.

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".

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

A rational number is any number that can be expressed as the fraction of two integers. The numerator is an integer, and the denominator is a non-zero integer. For example, 3/4 is a rational number, as is 5/1 (which equals 5).

The rational number 3/5 lies between consecutive integers -1 and 0.

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.

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.