Explain an error problem in each of the following:
a) 13/35=1/5, 27/73 = 2/3, 16/64 = 1/4
b) 4/5+2/3=6/8, 2/5+3/4=5/9, 7/8+1/3=8/11
c) 8 3/8-6 1/4=2 2/4, 5 3/8-2 2/3=3 1/5, 2 2/7-1 1/3=1 1/4
d) 2/3*3=6/9, 1/4*6=6/24, 4/5*2=8/10
What fun!
a) You can cancel out numbers that are common to the top and bottom of a fraction... but you can't cancel individual digits!
b) These examples suggest that a/b + c/d = (a+c)/(b+d). Wrong!
c) These examples suggest that {a+(b/c)} - {d+(e/f)} = {(a-d) + (b-e)/(f-d)}. Also wrong!
d) These examples suggest that a*(b/c) = (a*b)/(a*c). Wrong yet again!
Oops: I've got the last bit of the explanation of c) incorrect, but I'm sure you can correct it.
a) In these examples, the error problem lies in simplification. When simplifying fractions, we need to express them in their simplest form. However, each of the given fractions is not in its simplest form.
For instance, in the first example, 13/35 should be simplified to its simplest form. To do this, we find the greatest common divisor (GCD) of both 13 and 35, which is 1. Dividing both the numerator and the denominator by 1 gives us the simplest form, which is 13/35.
Similarly, the second and third examples also have fractions that can be simplified further to their simplest form. It is important to simplify fractions correctly to ensure accurate calculations.
b) The error problem in these examples is also related to simplification. When adding fractions, we need to have a common denominator. However, the given fractions do not have a common denominator. As a result, the addition is incorrect.
In the first example, 4/5 + 2/3, we need to find a common denominator. The common denominator is 15. Converting both fractions to have the common denominator, we get 12/15 + 10/15, which equals 22/15. Therefore, the correct answer is 22/15, not 6/8.
Similarly, the second and third examples require finding a common denominator and performing the addition correctly. It is important to understand the concept of finding common denominators when adding fractions.
c) In these examples, the error problem lies in the subtraction of mixed numbers. When subtracting mixed numbers, we need to borrow/regroup if the fraction part of the second number is larger than the fraction part of the first number. However, in the given examples, this regrouping is not done correctly.
For example, in the first example, 8 3/8 - 6 1/4, we need to regroup because 1/4 is larger than 3/8. Regrouping 8 as 7 16/8, the subtraction becomes (7 16/8) - (6 1/4). Simplifying further, we get 1 8/8, which is equal to 1.
Similarly, the second and third examples require correct regrouping to obtain the correct answer. It is important to carefully subtract the mixed numbers, ensuring the proper regrouping is done when necessary.
d) The error problem in these examples is related to multiplication. When multiplying fractions, we need to multiply both the numerators and the denominators. However, it appears that only the numerators or denominators are being multiplied in these examples.
For instance, in the first example, 2/3 * 3, we need to multiply the numerator (2) with the numerator of 3 (which is 3) and the denominator (3) with the denominator of 3 (which is 1). This gives us (2 * 3) / (3 * 1) = 6/3 = 2. Therefore, the correct answer is 2, not 6/9.
Similarly, the second and third examples require multiplying the numerators and denominators correctly to obtain the correct answer. It is essential to remember the rule of multiplying fractions and perform the multiplication accurately.