Word Problems
4A · Chapter 2 · Lesson 5 · MATH-4A-C02-L05
Learning goal. Represent unknown totals and differences in multistep contexts using labeled bars or equations.
Before you begin
Recall: 4A Ch1: million-range place value; Grade 3: column algorithms and comparison bars; Other Ways to Add and Subtract — Part 2. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
Calculate 865,195 − 190,193 by adding 100 to both quantities first. What answer do you get?
Show answer & reasoning
Equal adjustment preserves the difference: (865,295) − (190,293) = 675,002.
Calculate 377,352 − 212,127 by adding 10 to both quantities first. What answer do you get?
Show answer & reasoning
Equal adjustment preserves the difference: (377,362) − (212,137) = 165,225.
Calculate 464,757 − 191,918 by adding 100 to both quantities first. What answer do you get?
Show answer & reasoning
Equal adjustment preserves the difference: (464,857) − (192,018) = 272,839.
Key vocabulary
- regroup
- exchange equal-value units
- compensation
- temporary adjustment corrected afterward
- sum
- result of addition
- difference
- result of subtraction
Understand the concept
Multistep word problems become easier when every quantity has a clear label. First identify the whole and its parts; then find which quantity is unknown. An additive comparison bar shows that two amounts can share equal portions plus an extra piece. Write an equation for the same relationships and check that your result satisfies the original story.
Model it: part-whole bars and labeled comparison bars. Explain what every part of the model represents, then write a matching equation.
Fully worked examples
Read the problem first and predict the answer. Then study how each quantity is represented and calculated.
Worked example 1
A library had 250,287 books, received 10,500 more, then lent 5,250. How many books are there now?
First add 260,787. Then subtract 5,250 to obtain 255,537.
Check your understanding 1
A library had 647,798 books, received 161,469 more, then lent 80,734. How many books are there now?
Show answer & reasoning
First add 809,267. Then subtract 80,734 to obtain 728,533.
Worked example 2
A library had 681,882 books, received 196,897 more, then lent 98,448. How many books are there now?
First add 878,779. Then subtract 98,448 to obtain 780,331.
Check your understanding 2
A library had 687,319 books, received 194,929 more, then lent 97,464. How many books are there now?
Show answer & reasoning
First add 882,248. Then subtract 97,464 to obtain 784,784.
Worked example 3
A library had 427,427 books, received 23,986 more, then lent 11,993. How many books are there now?
First add 451,413. Then subtract 11,993 to obtain 439,420.
Check your understanding 3
A library had 651,928 books, received 65,980 more, then lent 32,990. How many books are there now?
Show answer & reasoning
First add 717,908. Then subtract 32,990 to obtain 684,918.
Guided practice
A library had 147,096 books, received 62,116 more, then lent 31,058. How many books are there now?
Show answer & reasoning
First add 209,212. Then subtract 31,058 to obtain 178,154.
A library had 406,367 books, received 35,544 more, then lent 17,772. How many books are there now?
Show answer & reasoning
First add 441,911. Then subtract 17,772 to obtain 424,139.
A library had 27,727 books, received 67,608 more, then lent 33,804. How many books are there now?
Show answer & reasoning
First add 95,335. Then subtract 33,804 to obtain 61,531.
Alex and Bea have 614,755 points together. Bea has 1,345 more than Alex. How many does Alex have?
Show answer & reasoning
Remove the extra 1,345 from the whole and split the rest into two equal parts: (614,755 − 1,345) ÷ 2 = 306,705.
Alex and Bea have 275,787 points together. Bea has 4,769 more than Alex. How many does Alex have?
Show answer & reasoning
Remove the extra 4,769 from the whole and split the rest into two equal parts: (275,787 − 4,769) ÷ 2 = 135,509.
Independent practice
A library had 408,395 books, received 128,104 more, then lent 64,052. How many books are there now?
Show answer & reasoning
First add 536,499. Then subtract 64,052 to obtain 472,447.
A library had 530,490 books, received 203,419 more, then lent 101,709. How many books are there now?
Show answer & reasoning
First add 733,909. Then subtract 101,709 to obtain 632,200.
Alex and Bea have 377,323 points together. Bea has 1,731 more than Alex. How many does Alex have?
Show answer & reasoning
Remove the extra 1,731 from the whole and split the rest into two equal parts: (377,323 − 1,731) ÷ 2 = 187,796.
Alex and Bea have 1,226,346 points together. Bea has 4,416 more than Alex. How many does Alex have?
Show answer & reasoning
Remove the extra 4,416 from the whole and split the rest into two equal parts: (1,226,346 − 4,416) ÷ 2 = 610,965.
Alex and Bea have 543,144 points together. Bea has 3,482 more than Alex. How many does Alex have?
Show answer & reasoning
Remove the extra 3,482 from the whole and split the rest into two equal parts: (543,144 − 3,482) ÷ 2 = 269,831.
A library had 694,134 books, received 141,434 more, then lent 70,717. How many books are there now?
Show answer & reasoning
First add 835,568. Then subtract 70,717 to obtain 764,851.
A library had 621,421 books, received 101,816 more, then lent 50,908. How many books are there now?
Show answer & reasoning
First add 723,237. Then subtract 50,908 to obtain 672,329.
A library had 109,843 books, received 142,772 more, then lent 71,386. How many books are there now?
Show answer & reasoning
First add 252,615. Then subtract 71,386 to obtain 181,229.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
Two classes collect 11809 cans. Class B collects 2483 more than Class A. A student divides the total by 2 without removing the difference. Explain the mistake and find Class A.
Show answer & reasoning
Remove the extra 2483, then split the remaining 9326 equally: (11809 − 2483) ÷ 2 = 4663.
Alex and Bea have 778,003 points together. Bea has 3,531 more than Alex. How many does Alex have?
Show answer & reasoning
Remove the extra 3,531 from the whole and split the rest into two equal parts: (778,003 − 3,531) ÷ 2 = 387,236.
Exit check
Alex and Bea have 450,642 points together. Bea has 5,878 more than Alex. How many does Alex have?
Show answer & reasoning
Remove the extra 5,878 from the whole and split the rest into two equal parts: (450,642 − 5,878) ÷ 2 = 222,382.
Alex and Bea have 1,104,605 points together. Bea has 1,315 more than Alex. How many does Alex have?
Show answer & reasoning
Remove the extra 1,315 from the whole and split the rest into two equal parts: (1,104,605 − 1,315) ÷ 2 = 551,645.
Alex and Bea have 634,710 points together. Bea has 1,332 more than Alex. How many does Alex have?
Show answer & reasoning
Remove the extra 1,332 from the whole and split the rest into two equal parts: (634,710 − 1,332) ÷ 2 = 316,689.