Let's solve both problems step by step.
---
Problem 6:
A group of tourists can use buses of 40 or buses of 60 so that all the seats are taken. Find the smallest number of tourists in the group.
####
Understanding the Problem:
We are told that:
- The group can be divided evenly into buses of
40 seats, with
no empty seats.
- The group can also be divided evenly into buses of
60 seats, again with
no empty seats.
- We need to find the
smallest possible number of tourists that satisfies both conditions.
This means we're looking for the
smallest number that is divisible by both 40 and 60.
That is, we need to find the
Least Common Multiple (LCM) of 40 and 60.
---
####
Step 1: Prime Factorization
- $ 40 = 2^3 \times 5 $
- $ 60 = 2^2 \times 3 \times 5 $
####
Step 2: Take the highest powers of all primes
- $ 2^3 $ (from 40)
- $ 3^1 $ (from 60)
- $ 5^1 $ (common)
So,
$$
\text{LCM} = 2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120
$$
---
####
✔ Answer to Problem 6:
The smallest number of tourists is
120.
> ✔️ Because:
> - $ 120 \div 40 = 3 $ buses (exact)
> - $ 120 \div 60 = 2 $ buses (exact)
---
Problem 8:
Fay and Ron both went to the gym today. Fay goes to the gym every 5 days and Ron goes to the gym every 7 days. How many days will it be until Fay and Ron both go to the gym on the same day again?
####
Understanding the Problem:
- Fay goes every
5 days: Day 0, 5, 10, 15, 20, ...
- Ron goes every
7 days: Day 0, 7, 14, 21, ...
- We want to know when their schedules
next coincide after today.
This is asking for the
Least Common Multiple (LCM) of 5 and 7.
---
####
Step 1: Prime Factorization
- $ 5 = 5 $
- $ 7 = 7 $
Since both are prime and different, the LCM is:
$$
\text{LCM} = 5 \times 7 = 35
$$
---
####
✔ Answer to Problem 8:
They will both go to the gym on the same day again in
35 days.
> ✔️ Check:
> - Fay: 0, 5, 10, 15, 20, 25, 30,
35
> - Ron: 0, 7, 14, 21, 28,
35
> → They meet again on day 35.
---
✔ Final Answers:
-
Problem 6: $ \boxed{120} $
-
Problem 8: $ \boxed{35} $
These are solved using
LCM concepts, which is the key idea in such word problems involving repeated cycles or equal division.
Parent Tip: Review the logic above to help your child master the concept of lcm word problems worksheet.
