Login
Premium
Home Question Bank Math Arithmetic
Question
upstudy study bank question image url

Write each number in the form \( \frac{a}{b} \) where \( a \) and \( b \in \mathbb{Z} \) and \( a \) and \( b \) have no common factors. \( \begin{array}{ll}\text { a) } \frac{2}{3} \times \frac{6}{5} & \text { b) } \frac{-15}{8} \times \frac{12}{10} \\ \text { c) }-1 \frac{3}{4} \times \frac{8}{9} & \text { d) } 2 \frac{4}{5} \times \frac{-10}{12} \\ \text { e) } \frac{4}{6} \div \frac{4}{3} & \text { f) } \frac{-6}{25} \div \frac{5}{12} \\ \text { g) } \frac{1}{2}+\frac{1}{3}+\frac{1}{4} & \text { h) } 2 \frac{2}{3}+1 \frac{1}{4}-3 \frac{1}{5} \\ \text { i) } \frac{2}{3} \times \frac{3}{4} \div \frac{5}{4}+\frac{4}{5} \div \frac{1}{2} & \text { j) } \frac{2}{3} \div \frac{3}{4} \div \frac{5}{4}+\frac{4}{5} \div 2 \\ \text { k) } \frac{2}{3} \div\left(\frac{3}{4} \div \frac{5}{4}\right)-\frac{4}{3} \div \frac{6}{2} & \text { 1) } \frac{3}{2} \div \frac{3}{4} \times\left(\frac{5}{4}-\frac{4}{5}\right) \div \frac{1}{2}\end{array} \)

Ask by Hobbs Mckenzie. in South Africa
Jan 21,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

Here are the simplified answers for each expression: a) \( \frac{4}{5} \) b) \( -\frac{9}{4} \) c) \( -\frac{14}{9} \) d) \( -\frac{7}{3} \) e) \( \frac{1}{2} \) f) \( -\frac{72}{125} \) g) \( \frac{13}{12} \) h) \( \frac{43}{60} \) i) \( 2 \) or \( \frac{2}{1} \) j) \( \frac{10}{9} \) k) \( \frac{2}{3} \) 1) \( \frac{9}{5} \)

Solution

Sure! Let's simplify each expression and write the result in the form \( \frac{a}{b} \) where \( a \) and \( b \) are integers with no common factors. ### a) \( \frac{2}{3} \times \frac{6}{5} \) \[ \frac{2}{3} \times \frac{6}{5} = \frac{2 \times 6}{3 \times 5} = \frac{12}{15} = \frac{4}{5} \] **Answer:** \( \frac{4}{5} \) --- ### b) \( \frac{-15}{8} \times \frac{12}{10} \) \[ \frac{-15}{8} \times \frac{12}{10} = \frac{-15 \times 12}{8 \times 10} = \frac{-180}{80} = \frac{-9}{4} \] **Answer:** \( -\frac{9}{4} \) --- ### c) \( -1 \frac{3}{4} \times \frac{8}{9} \) First, convert the mixed number to an improper fraction: \[ -1 \frac{3}{4} = -\frac{7}{4} \] Then multiply: \[ -\frac{7}{4} \times \frac{8}{9} = \frac{-56}{36} = \frac{-14}{9} \] **Answer:** \( -\frac{14}{9} \) --- ### d) \( 2 \frac{4}{5} \times \frac{-10}{12} \) Convert the mixed number to an improper fraction: \[ 2 \frac{4}{5} = \frac{14}{5} \] Then multiply: \[ \frac{14}{5} \times \frac{-10}{12} = \frac{-140}{60} = \frac{-7}{3} \] **Answer:** \( -\frac{7}{3} \) --- ### e) \( \frac{4}{6} \div \frac{4}{3} \) Division by a fraction is equivalent to multiplication by its reciprocal: \[ \frac{4}{6} \div \frac{4}{3} = \frac{4}{6} \times \frac{3}{4} = \frac{12}{24} = \frac{1}{2} \] **Answer:** \( \frac{1}{2} \) --- ### f) \( \frac{-6}{25} \div \frac{5}{12} \) \[ \frac{-6}{25} \div \frac{5}{12} = \frac{-6}{25} \times \frac{12}{5} = \frac{-72}{125} \] **Answer:** \( -\frac{72}{125} \) --- ### g) \( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \) Find a common denominator (12): \[ \frac{1}{2} = \frac{6}{12}, \quad \frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12} \] Add them: \[ \frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{13}{12} \] **Answer:** \( \frac{13}{12} \) --- ### h) \( 2 \frac{2}{3} + 1 \frac{1}{4} - 3 \frac{1}{5} \) Convert all mixed numbers to improper fractions: \[ 2 \frac{2}{3} = \frac{8}{3}, \quad 1 \frac{1}{4} = \frac{5}{4}, \quad 3 \frac{1}{5} = \frac{16}{5} \] Find a common denominator (60): \[ \frac{8}{3} = \frac{160}{60}, \quad \frac{5}{4} = \frac{75}{60}, \quad \frac{16}{5} = \frac{192}{60} \] Combine them: \[ \frac{160}{60} + \frac{75}{60} - \frac{192}{60} = \frac{43}{60} \] **Answer:** \( \frac{43}{60} \) --- ### i) \( \frac{2}{3} \times \frac{3}{4} \div \frac{5}{4} + \frac{4}{5} \div \frac{1}{2} \) Step-by-step calculation: \[ \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2} \] \[ \frac{1}{2} \div \frac{5}{4} = \frac{1}{2} \times \frac{4}{5} = \frac{4}{10} = \frac{2}{5} \] \[ \frac{4}{5} \div \frac{1}{2} = \frac{4}{5} \times \frac{2}{1} = \frac{8}{5} \] \[ \frac{2}{5} + \frac{8}{5} = \frac{10}{5} = 2 \] **Answer:** \( 2 \) or \( \frac{2}{1} \) --- ### j) \( \frac{2}{3} \div \frac{3}{4} \div \frac{5}{4} + \frac{4}{5} \div 2 \) Step-by-step calculation: \[ \frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9} \] \[ \frac{8}{9} \div \frac{5}{4} = \frac{8}{9} \times \frac{4}{5} = \frac{32}{45} \] \[ \frac{4}{5} \div 2 = \frac{4}{5} \times \frac{1}{2} = \frac{4}{10} = \frac{2}{5} \] \[ \frac{32}{45} + \frac{2}{5} = \frac{32}{45} + \frac{18}{45} = \frac{50}{45} = \frac{10}{9} \] **Answer:** \( \frac{10}{9} \) --- ### k) \( \frac{2}{3} \div \left( \frac{3}{4} \div \frac{5}{4} \right) - \frac{4}{3} \div \frac{6}{2} \) Step-by-step calculation: \[ \frac{3}{4} \div \frac{5}{4} = \frac{3}{4} \times \frac{4}{5} = \frac{12}{20} = \frac{3}{5} \] \[ \frac{2}{3} \div \frac{3}{5} = \frac{2}{3} \times \frac{5}{3} = \frac{10}{9} \] \[ \frac{6}{2} = 3, \quad \frac{4}{3} \div 3 = \frac{4}{3} \times \frac{1}{3} = \frac{4}{9} \] \[ \frac{10}{9} - \frac{4}{9} = \frac{6}{9} = \frac{2}{3} \] **Answer:** \( \frac{2}{3} \) --- ### 1) \( \frac{3}{2} \div \frac{3}{4} \times \left( \frac{5}{4} - \frac{4}{5} \right) \div \frac{1}{2} \) Step-by-step calculation: \[ \frac{3}{2} \div \frac{3}{4} = \frac{3}{2} \times \frac{4}{3} = 2 \] \[ \frac{5}{4} - \frac{4}{5} = \frac{25}{20} - \frac{16}{20} = \frac{9}{20} \] \[ 2 \times \frac{9}{20} = \frac{18}{20} = \frac{9}{10} \] \[ \frac{9}{10} \div \frac{1}{2} = \frac{9}{10} \times \frac{2}{1} = \frac{18}{10} = \frac{9}{5} \] **Answer:** \( \frac{9}{5} \) ---

Reviewed and approved by the UpStudy tutoring team

error msg
Explain
Simplify this solution

Extra Insights

Let's simplify each expression step by step: a) \[ \frac{2}{3} \times \frac{6}{5} = \frac{2 \times 6}{3 \times 5} = \frac{12}{15} = \frac{4}{5} \] b) \[ \frac{-15}{8} \times \frac{12}{10} = \frac{-15 \times 12}{8 \times 10} = \frac{-180}{80} = \frac{-9}{4} \quad (\text{after dividing by } 20) \] c) \[ -1 \frac{3}{4} \times \frac{8}{9} = -\frac{7}{4} \times \frac{8}{9} = \frac{-56}{36} = \frac{-14}{9} \quad (\text{after dividing by } 4) \] d) \[ 2 \frac{4}{5} \times \frac{-10}{12} = \frac{14}{5} \times \frac{-10}{12} = \frac{-140}{60} = \frac{-7}{3} \quad (\text{after dividing by } 20) \] e) \[ \frac{4}{6} \div \frac{4}{3} = \frac{4}{6} \times \frac{3}{4} = \frac{3}{6} = \frac{1}{2} \] f) \[ \frac{-6}{25} \div \frac{5}{12} = \frac{-6}{25} \times \frac{12}{5} = \frac{-72}{125} \] g) \[ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{13}{12} \] h) \[ 2 \frac{2}{3} + 1 \frac{1}{4} - 3 \frac{1}{5} = \frac{8}{3} + \frac{5}{4} - \frac{16}{5} = \frac{40}{15} + \frac{15}{15} - \frac{48}{15} = \frac{7}{15} \] i) \[ \frac{2}{3} \times \frac{3}{4} \div \frac{5}{4} + \frac{4}{5} \div \frac{1}{2} = \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} + \frac{4}{5} \times 2 = \frac{24}{60} + \frac{8}{5} = \frac{2}{5} + \frac{8}{5} = \frac{10}{5} = 2 \] j) \[ \frac{2}{3} \div \frac{3}{4} \div \frac{5}{4} + \frac{4}{5} \div 2 = \frac{2}{3} \times \frac{4}{3} \times \frac{4}{5} + \frac{4}{5} \times \frac{1}{2} = \frac{32}{45} + \frac{4}{10} = \frac{32}{45} + \frac{18}{45} = \frac{50}{45} = \frac{10}{9} \] k) \[ \frac{2}{3} \div \left( \frac{3}{4} \div \frac{5}{4} \right) - \frac{4}{3} \div \frac{6}{2} = \frac{2}{3} \div \left( \frac{3}{5} \right) - \frac{4}{3} \times \frac{1}{3} = \frac{10}{9} - \frac{4}{9} = \frac{6}{9} = \frac{2}{3} \] 1) \[ \frac{3}{2} \div \frac{3}{4} \times \left( \frac{5}{4} - \frac{4}{5} \right) \div \frac{1}{2} = \frac{3}{2} \times \frac{4}{3} \times \left(

Related Questions

a. Use each digit once. Make the smallest 3-digit number: Use each digit once. Make the largest 3-digit number. You can use a digit twice, Make the smallest 3-digit number:
Arithmetic South Africa Jan 22, 2025
(40) Jamaal writes \( \frac{1}{8} \) page in \( \frac{1}{4} \) of a minute. How much time will it take Jamaal to write a full page?
Arithmetic United States Jan 22, 2025
2. Start at 0 and count in these intervals. Write down the numbers. a) Count in 50 s up to 500 b) Count in 200s up to 1000
Arithmetic South Africa Jan 22, 2025
(13) Adding the identity element for addition to a number leaves the number unchanged. (14) Multiplying a number by the identity element for addition leaves the number (15) Snchanged. Subtracting the identity clement for addition from a number leaves the sumber (16) Multiplying a number by the identity element for multiplication leaves the number unchanged.
Arithmetic South Africa Jan 22, 2025
Celeste Phillips Calculating Square Footage The first step I took was to completely watch the "How to Calculate Square Footage" video and take notes. I did the example and made sure I fully understood it before moving on to the discussion word problem. The picture showed two rooms, one with a tile floor, and one with a carpeted floor. The word problem stated that only the total cost of the carpeted floor was needed. The length and width were both 20 ft , so I multiplied that together and got a square footage of 400. The price of carpeting per square foot was \( \$ 2.25 \), so I multiplied that by 400 and got a total price of \( \$ 900 \) for the carpeted floor.
Arithmetic United States Jan 22, 2025

Latest Arithmetic Questions

Celeste Phillips Calculating Square Footage The first step I took was to completely watch the "How to Calculate Square Footage" video and take notes. I did the example and made sure I fully understood it before moving on to the discussion word problem. The picture showed two rooms, one with a tile floor, and one with a carpeted floor. The word problem stated that only the total cost of the carpeted floor was needed. The length and width were both 20 ft , so I multiplied that together and got a square footage of 400. The price of carpeting per square foot was \( \$ 2.25 \), so I multiplied that by 400 and got a total price of \( \$ 900 \) for the carpeted floor.
Arithmetic United States Jan 22, 2025
Sarah went apple picking and had 4 baskets with 9 apples in each basket. Then she picked 3 more apples. Write an expression to show how many apples Sarah had. Then evaluate the expression. ach. food. face went to a taco shop for lunch and ordered 3 tacos for \( \$ 2 \) each. He paid with a \( \$ 10 \) bill. Write an expression to show how much change he should get back from the cashier. Then evaluate the expression.
Arithmetic United States Jan 22, 2025
(40) Jamaal writes \( \frac{1}{8} \) page in \( \frac{1}{4} \) of a minute. How much time will it take Jamaal to write a full page?
Arithmetic United States Jan 22, 2025
(13) Adding the identity element for addition to a number leaves the number unchanged. (14) Multiplying a number by the identity element for addition leaves the number (15) Snchanged. Subtracting the identity clement for addition from a number leaves the sumber (16) Multiplying a number by the identity element for multiplication leaves the number unchanged.
Arithmetic South Africa Jan 22, 2025
Rewrite each statement as a numerical expression. Add 5 and 6 then add 12 . subtract 8 from 32 then divide by 6 . Find the product of 2 and 7 then add 10 . Find 3 times as many as 4 times 2 .
Arithmetic United States Jan 22, 2025
Prev Next
Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy