The topic Operations with rational numbers used to appear under the 7th grade (U.S.) Math Mission. This exercise intensifies knowledge of operations on rational numbers with several different types of problems.

Types of Problems

There are four types of problems in this exercise:

  1. Select the expressions equivalent to: This problem provides a rational number. The student is asked to select all of the expressions from a multiple select list that are equivalent to the given number.

    Select the expressions equivalent to

  2. Given numbers, tell what is true: This problem defines two variables at the beginning of the problem. It then asks the student to select the statements that are accurate from among several operations on the variables and comparison with neutral zero.

    Given numbers, tell what is true

  3. Example or counterexample: This problem provides a statement made by someone and a chart of possible examples. The student is to select whether each example is true, or constitutes a counterexample.

    Example or counterexample

  4. Compare to zero: This problem provides several arithmetic expressions with rational numbers. The student is asked to select which are positive and which are negative.

    Compare to zero


Although this exercise is about operations on real numbers, it also involves comparison of numbers to zero, and basic knowledge regarding examples versus counterexamples from logic.

  1. A counterexample is an example that witnesses a particular statement is false.
  2. Recognizing the magnitude of the fractions can assist with speed on Given numbers, tell what is true.
  3. On Example or counterexample it is useful to remember that multiplying by a number greater than one makes numbers bigger and multiplying by a number smaller than one makes them smaller.
  4. Tips would be greatly appreciated for the first problem type.

Real-life Applications

  1. Taxes are a fraction of the income or the purchase amount, so if one spends or earns money, they're involved with fractions.
  2. If a pizza or bag of popcorn is shared, each recipient will get a fraction of it.
  3. Interest rates on loans and mortgages are fractions.
  4. Interest on a savings account is a fraction.
  5. Taxes on gasoline are fractions of the amount purchased.

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