All the solutions provided in McGraw Hill My Math Grade 4 Answer Key PDF Chapter 9 Lesson 5 Problem-Solving Investigation: Work Backward will give you a clear idea of the concepts.
McGraw-Hill My Math Grade 4 Answer Key Chapter 9 Lesson 5 Problem-Solving Investigation: Work Backward
Learn the Strategy
Nathan used some flour for a cake recipe. He used \(\frac{1}{4}\) of the bag of flour for a bread recipe. There is \(\frac{2}{4}\) of the bag left. What fraction of the bag did Nathan use for the cake?
1. Understand
What facts do you know?
Nathan used some flour for a cake recipe and ______________ of the bag for a bread recipe. He has ______________ of the bag left.
What do you need to find?
Find the fraction of the bag of flour that was used for the ______________.
2. Plan
I will work backward to solve the problem.
3. Solve
So, ________________ of the bag of flour was used for the cake.
4. Check
Does your answer make sense? Explain.
Answer:
Fraction of the bag Nathan used for the cake = \(\frac{1}{4}\)
My answer makes sense because the values are matching and completes the problem.
Explanation:
Nathan used some flour for a cake recipe.
Quantity of the bag of flour for a bread recipe he used = \(\frac{1}{4}\)
Quantity of the bag of flour left = \(\frac{2}{4}\)
Fraction of the bag Nathan used for the cake = Quantity of the bag of flour left – Quantity of the bag of flour for a bread recipe he used
= \(\frac{2}{4}\) – \(\frac{1}{4}\)
= [(2 – 1) ÷ 4]
= \(\frac{1}{4}\)
Check:
Quantity of the bag of flour for a bread recipe he used + Fraction of the bag Nathan used for the cake + Quantity of the bag of flour left
= \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{2}{4}\)
= [(1 + 1) ÷ 4] + \(\frac{2}{4}\)
= \(\frac{2}{4}\) + \(\frac{2}{4}\)
= [(2 + 2) ÷ 4]
= \(\frac{4}{4}\)
= 1.
Practice the Strategy
Lily, Noah, and Kaylee are sharing pizza. Noah ate \(\frac{2}{6}\) of the pizza. Kaylee ate \(\frac{1}{6}\) of the pizza. There is \(\frac{2}{6}\)of the pizza left. What fraction of the pizza did Lily eat?
1. Understand
What facts do you know?
What do you need to find?
2. Plan
3. Solve
4. Check
Does your answer make sense? Explain.
Answer:
Fraction of the pizza Lily eat = \(\frac{1}{6}\).
My answer makes sense because the values are matching and completes the problem.
Explanation:
Quantity of of the pizza Noah ate = \(\frac{2}{6}\)
Quantity of of the pizza Kaylee ate = \(\frac{1}{6}\)
Quantity of of the pizza left = \(\frac{2}{6}\)
Fraction of the pizza Lily eat = (Quantity of of the pizza Noah ate + Quantity of of the pizza Kaylee ate) – Quantity of of the pizza left
= (\(\frac{2}{6}\) + \(\frac{1}{6}\)) – \(\frac{2}{6}\)
= [(2 + 1) ÷ 6] – \(\frac{2}{6}\)
= \(\frac{3}{6}\) – \(\frac{2}{6}\)
= [(3 – 2) ÷ 6]
= \(\frac{1}{6}\)
Check:
Quantity of of the pizza Noah ate + Quantity of of the pizza Kaylee ate + Fraction of the pizza Lily eat + Quantity of of the pizza left
= \(\frac{2}{6}\) + \(\frac{1}{6}\) + \(\frac{2}{6}\) + \(\frac{1}{6}\)
= [(2 + 1) ÷ 6] + [(2 + 1) ÷ 6]
= \(\frac{3}{6}\) + \(\frac{3}{6}\)
= [(3 + 3) ÷ 6]
= \(\frac{6}{6}\)
= 1.
Apply the Strategy
Solve each problem by working backward.
Question 1.
Mathematical PRACTICE Make Sense of Problems Chloe did some of her homework before dinner She did \(\frac{2}{6}\) of her homework after dinner. She has \(\frac{1}{6}\) of her homework left. What fraction of her homework did Chloe do before her dinner? Write in simplest form.
Answer:
Fraction of her homework Chloe do before her dinner = \(\frac{1}{6}\)
Explanation:
Portion of of her homework after dinner she did = \(\frac{2}{6}\)
Portion of of her homework she has left = \(\frac{1}{6}\)
Fraction of her homework Chloe do before her dinner = Portion of of her homework after dinner she did – Portion of of her homework she has left
= \(\frac{2}{6}\) – \(\frac{1}{6}\)
= [(2 – 1) ÷ 6]
= \(\frac{1}{6}\)
Question 2.
There were 12 goals scored during the game. Team A scored \(\frac{8}{12}\) of the goals. Team B scored 2 goals during the first half of the game. What fraction of the goals did Team B score during the second half of the game? Write in simplest form.
Answer:
Fraction of the goals did Team B score during the second half of the game = \(\frac{28}{3}\)
Explanation:
Number of goals scored during the game = 12.
Number of goals scored by Team A = \(\frac{8}{12}\)
Number of goals scored by Team B during the first half of the game = 2.
Fraction of the goals did Team B score during the second half of the game = Number of goals scored during the game – Number of goals scored by Team A – Number of goals scored by Team B during the first half of the game)
= 12 – (\(\frac{8}{12}\) + 2)
= 12 – [(8 + 24) ÷ 12]
= 12 – (32 ÷ 12)
= [(12 × 12) – 32] ÷ 12
= (144 – 32) ÷ 12
= 112 ÷ 12 or \(\frac{112}{12}\) ÷ \(\frac{4}{4}\)
= \(\frac{28}{3}\)
Question 3.
Brandi and her mom are at a pet store. The pet store has 12 reptiles. Of the reptiles, \(\frac{5}{12}\) are turtles, \(\frac{2}{12}\) are snakes, and the rest are lizards. What fraction of the reptiles is lizards?
Answer:
Fraction of the reptiles is lizards = \(\frac{137}{12}\)
Explanation:
Number of reptiles the pet store has = 12.
Number of turtles = \(\frac{5}{12}\)
Number of snakes = \(\frac{2}{12}\)
Fraction of the reptiles is lizards = Number of reptiles the pet store has – (Number of turtles + Number of snakes)
= 12 – ( \(\frac{5}{12}\) + \(\frac{2}{12}\))
= 12 – [(5 + 2) ÷ 12]
= 12 – \(\frac{7}{12}\)
= {[(12 × 12) – 7] ÷ 12}
= [144 – 7) ÷ 12
= 137 ÷ 12 or \(\frac{137}{12}\)
Review the Strategies
Use any strategy to solve each problem.
- Work backward.
- Use logical reasoning.
- Look for a pattern.
- Make a model.
Question 4.
There are 16 books on a shelf. Four-sixteenths of the books are about animals. Two are adventure. The rest are mystery. How many are mystery books?
Answer:
Number of books are mystery = \(\frac{55}{4}\)
Explanation:
Number of books on a shelf = 16.
Number of books are about animals = Four-sixteenths or \(\frac{4}{16}\)
Number of books are adventure = 2.
Number of books are mystery = Number of books on a shelf – (Number of books are about animals + Number of books are adventure)
= 16 – (\(\frac{4}{16}\) + 2)
= 16 – [4 + (2 × 16) ÷ 16]
= 16 – [(4 + 32) ÷ 16]
= 16 – (36 ÷ 16)
= {[(16 × 16) – 36] ÷ 16}
= (256 – 36) ÷ 16
= 220 ÷ 16 or \(\frac{220}{16}\) ÷ \(\frac{4}{4}\)
= \(\frac{55}{4}\)
Question 5.
There are 10 pieces of chalk. Two-tenths of the chalk is pink. One piece is blue. The rest are white. How many pieces of chalk are white?
Answer:
Number of pieces of chalk are white = \(\frac{44}{5}\)
Explanation:
Number of pieces of chalk = 10.
Number of pieces of chalk are pink = Two-tenths or \(\frac{2}{10}\)
Number of pieces of chalk are blue = 1.
Number of pieces of chalk are white = Number of pieces of chalk – (Number of pieces of chalk are pink + Number of pieces of chalk are blue)
= 10 – (\(\frac{2}{10}\) + 1)
= 10 – {[(2 + (1 × 10)] ÷ 10}
= 10 – [(2 + 10) ÷ 10]
= 10 – (12 ÷ 10)
= {[(10 × 10) – 12] ÷ 10}
= [(100 – 12) ÷ 10]
= (88 ÷ 10) ÷ \(\frac{2}{2}\)
= \(\frac{44}{5}\)
Question 6.
Giselle played with some friends on Monday. She played with 2 times as many friends on Wednesday. This was 4 more than on Friday. On Friday, she played with 4 friends. How many did she play with on Monday?
Answer:
Number of friends she played on Monday = 4.
Explanation:
Number of times as many friends she played on Wednesday = 2.
This was 4 more than on Friday.
=> Number of friends she played on Friday = 4.
Number of friends she play with on Wednesday = 4 × Number of friends she played on Friday
= 4 × 2
= 8.
Number of friends she played on Monday = Number of friends she play with on Wednesday – Number of friends she played on Friday
= 8 – 4
= 4.
Question 7.
Mathematical PRACTICE Use Math Tools Mrs. Vargas is making costumes for a play. She needs 3 buttons for each costume. Complete the table to find how many buttons she will need for 22 costumes.
Answer:
Explanation:
Number of buttons for each costume she needs = 3 .
Number of costumes = 22.
Number of buttons she needs for 22 costumes = Number of buttons for each costume she needs × Number of costumes
= 22 × 3
= 66.
Number of buttons she needs for 21 costumes = Number of buttons for each costume she needs × Number of costumes
= 21 × 3
= 63.
McGraw Hill My Math Grade 4 Chapter 9 Lesson 5 My Homework Answer Key
Problem Solving
Solve each problem by working backward.
Question 1.
Marla, Jamie, and Sarah have each led their book club’s monthly meeting. Jamie has led \(\frac{2}{6}\) of the meetings, and Sarah has led \(\frac{1}{6}\) of the meetings. What fraction of the meetings has Marla led?
Answer:
Fraction of the meetings has Marla led = \(\frac{1}{6}\)
Explanation:
Number of books Jamie has led of the meetings = \(\frac{2}{6}\)
Number of books Sarah has led of the meetings = \(\frac{1}{6}\)
Fraction of the meetings has Marla led = Number of books Jamie has led of the meetings – Number of books Sarah has led of the meetings
= \(\frac{2}{6}\) – \(\frac{1}{6}\)
= [(2 – 1) ÷ 6]
= \(\frac{1}{6}\)
Question 2.
Suzanne dropped her penny jar. She found some of the pennies, but some are still missing. She found \(\frac{6}{10}\) of the pennies on the rug. She found \(\frac{3}{10}\) of the pennies on the couch. What fraction of pennies is still missing?
Answer:
Fraction of pennies is still missing = \(\frac{3}{10}\)
Explanation:
Number of the pennies on the rug she found = \(\frac{6}{10}\)
Number of the pennies on the couch she found = \(\frac{3}{10}\)
Fraction of pennies is still missing = Number of the pennies on the rug she found – Number of the pennies on the couch she found
= \(\frac{6}{10}\) – \(\frac{3}{10}\)
= [(6 – 3) ÷ 10]
= \(\frac{3}{10}\)
Question 3.
Mathematical PRACTICE Use Number Sense Noah spent some of his allowance on Monday. He spent \(\frac{1}{6}\) of his allowance on Tuesday and \(\frac{3}{6}\) of it on Friday. Noah has none of his allowance money left. What fraction of his allowance did he spend on Monday?
Answer:
Amount of his allowance Noah spent on Monday = \(\frac{1}{3}\)
Explanation:
Amount of his allowance Noah spent on Tuesday = \(\frac{1}{6}\)
Amount of his allowance Noah spent on Friday = \(\frac{3}{6}\)
Amount of his allowance Noah spent on Monday = Amount of his allowance Noah spent on Friday – Amount of his allowance Noah spent on Tuesday
= \(\frac{3}{6}\) – \(\frac{1}{6}\)
= [(3 – 1) ÷ 6]
= \(\frac{2}{6}\) ÷ \(\frac{2}{2}\)
= \(\frac{1}{3}\)